Skip to main content

Full text of "Mathcad computer applications predicting antenna parameters from antenna physical dimensions and ground characteristics"

See other formats







NPSEC-93-012 



NAVAL POSTGRADUATE SCHOOL 

Monterey, California 




fathead Computer Applications 

Predicting Antenna Parameters from 

Antenna Physical Dimensions and 

Ground Characteristics 

by 

Donald D. Gerry 
n 

R. Clark Robertson 
June 1993 



PedDocs 

D 208.14/2 

NPS-EC-93-012 



Approved for public reslease, distribution unlimited. 

Prepared for: NAVMARINTCEN 
D1433 

4301 Suitland 
Washington D.C. 20395-5020 



6 oct 



Naval Postgraduate School 

Monterey, California 93943-5000 



Rear Admiral T. A. Mercer H. Schull 

Superintendent Provost 

This report was funded by NAVMARINTCEN. 
Reproduction of all or part of this report is authorized. 
This report was prepared by: _ 



1\ IRI FY KNfiX UBRftflY 



REPORT DOCUMENTATION PAGE 



wal pmv<3mmA7& schooi 




1. AGENCY USE ONLY (Leave blank) 



!. REPORT DATE 
Jun 93 



3. REPORT TYPE AND DATES COVERED 
Interim Dec 92 - Jun 93 



4. TITLE AND SUBTITLE 



MATHCAD COMPUTER APPLICATIONS PREDICTING ANTENNA 
PARAMETERS FROM ANTENNA PHYSICAL DIMENSIONS AND GROUND 
CHARACTERISTICS 



6. AUTHOR(S) 



Gerry, Donald D. and R. Clark Robertson 



5. FUNDING NUMBERS 



7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

Naval Postgraduate School 
Monterey, CA 93943-5002 



PERFORMING ORGANIZATION 
REPORT NUMBER 

NPSEC-93-012 



9. SPONSORING /MONITORING AGENCY NAME(S) AND ADDRESS(ES) 



10. SPONSORING /MONITORING 
AGENCY REPORT NUMBER 



NAVMARINTCEN 

DI433 

4301 Suit land 

Washington, D.C. 20395-5020 



11. SUPPLEMENTARY NOTES 



The views expressed in this report are those of the author and do not reflect the official policy or 
position of the Department of Defense or U.S. Government 



12a. DISTRIBUTION /AVAILABILITY STATEMENT 

Approved for public release; distribution is unlimited 



12b. DISTRIBUTION CODE 



13. ABSTRACT (Maximum 200 words) 



This report provides the documentation for a set of computer applications for the evaluation 
of antenna parameters. The applications are written for the Mathcad personal computer software for 
various antenna types listed in the thesis index. Antenna dimensions and, in some cases, ground 
parameters are the only required inputs for each application. No new antenna parameter equations were 
developed as a part of this research. 

The chapters of this thesis are intended to provide Mathcad antenna application users with the 
background information necessary to readily use and interpret the software for each antenna type. 
Appendices are provided with examples of each antenna application. Each application has an introductory 
paragraph and a table of required inputs. 

The Mathcad software provides various numerical outputs and performance predictions, as well 
as a graphical representation of radiation patterns in the far-field. Mathcad application results are 
consistent with the predictions of applicable publications, as well as other antenna numerical analysis 
programs. 



14. SUBJECT TERMS 

Radiation Pattern, Radiated Power, Directivity, Gain, Polarization, 
Efficiency, Effective Height/Area, Bandwidth, Wavelength, Effective 
Isotropic Radiated Power, Input Impedance, Reflection Coefficient 



15. NUMBER OF PAGES 
271 



16. PRICE CODE 



17. SECURITY CLASSIFICATION 
OF REPORT 

Unclassified 



18. SECURITY CLASSIFICATION 
OF THIS PAGE 

Unclassified 



19. SECURITY CLASSIFICATION 
OF ABSTRACT 

Unclassified 



20. LIMITATION OF ABSTRACT 

UL 



NSN 7540-01-280-5500 



Standard Form 298 (Rev 2-89) 

Prescribe by ANSI Std £39- 18 



XI 



ABSTRACT 

This report provides the documentation for a set of computer 
applications for the evaluation of antenna parameters. The 
applications are written for the Mathcad personal computer 
software for various antenna types listed in the thesis index. 
Antenna dimensions and, in some cases, ground parameters are the 
only required inputs for each application. No new antenna 
parameter equations were developed as a part of this research. 

The chapters of this thesis are intended to provide Mathcad 
antenna application users with the background information 
necessary to readily use and interpret the software for each 
antenna type. Appendices are provided with examples of each 
antenna application. Each application has an introductory 
paragraph and a table of required inputs. 

The Mathcad software provides various numerical outputs and 
performance predictions, as well as a graphical representation of 
radiation patterns in the far-field. Mathcad application results 
are consistent with the predictions of applicable publications, 
as well as other antenna numerical analysis programs. 



111 



TABLE OF CONTENTS 

I . BACKGROUND AND PURPOSE 1 

II. INTRODUCTION 2 

III. THE HELICAL ANTENNA 4 

IV . THE BEVERAGE ANTENNA 17 

V. THE LOOP ANTENNA 32 

A. THE ELECTRICALLY SMALL LOOP 34 

B. THE ELECTRICALLY LARGE LOOP 45 

VI . THE BEDSPRING ANTENNA 52 

VII . THE SPIRAL ANTENNA 67 

A . THE PLANAR SPIRAL ANTENNAS 67 

B. THE CONICAL SPIRAL ANTENNAS 8 

VIII. THE CONICAL HORN ANTENNA 93 

IX. THE PYRAMIDAL HORN ANTENNA 104 

X. REMARKS AND CONCLUSIONS 113 

APPENDIX A THE HELICAL ANTENNA , MATHCAD SOFTWARE HELIX. MCD. .. 115 
APPENDIX B THE BEVERAGE ANTENNA, MATHCAD SOFTWARE-BEVERAGE. MCD 

12 5 

APPENDIX C THE SMALL LOOP ANTENNA, MATHCAD SOFTWARE-SMLOOP.MCD 
135 

APPENDIX D THE LARGE LOOP ANTENNA, MATHCAD SOFTWARE-LGLOOP.MCD 
168 

APPENDIX E THE BEDSPRING ANTENNA, MATHCAD SOFTWARE-BEDSPRIN . MCD 
191 

APPENDIX F THE SPIRAL ANTENNA (EXACT METHOD) , MATHCAD SOFTWARE- 
SPIRAL. MCD 206 

iv 



APPENDIX G THE CONICAL HORN ANTENNA (EXACT METHOD) , MATHCAD 

SOFTWARE -HORN_CON . MCD 231 

APPENDIX H THE PYRAMIDAL HORN ANTENNA, MATHCAD SOFTWARE -HORN_ 

PYR . MCD 237 

REFERENCES 263 

INITIAL DISTRIBUTION LIST 265 



I. BACKGROUND AND PURPOSE 

This report and associated Mathcad computer software are 
submitted in partial fulfillment of the thesis requirements for 
the degree of Master of Science in Electrical Engineering from 
the Naval Postgraduate School in Monterey, CA. 

The thesis requirement was generated by a statement of work 
from the Naval Maritime Intelligence Center (NAVMARINTCEN) such 
that any IBM compatible personnel computer with MS-DOS version 
3.2 or higher and a math coprocessor could run Mathcad software 
applications to analyze the parameters of different antenna types 
requested by NAVMARINTCEN. 

Required user inputs to the applications are limited to 
antenna dimensions and ground data, although in some cases other 
data may be estimated to provide further insight into the 
antenna's performance. The Mathcad applications provide various 
performance predictions as well as a graphical representation of 
the antenna's far-field radiation pattern. The corresponding 
thesis chapter furnishes the application user all the necessary 
background information needed to interpret the program's formulas 
and displays, thereby allowing NAVMARINTCEN to interpret the 
capabilities and limitations of antennas of interest. 

Dietrich [Ref 1.] completed the first portion of this 
project. This thesis will be the second in a series of three 
reports intended to fulfill the NAVMARINTCEN statement of work. 



II. INTRODUCTION 

When a foreign country develops a new communications or 
radar system there are many reasons why various United States 
agencies may want to be appraised of the new eguipment's 
capabilities and limitations. Indeed, if the country is hostile 
to the United States, the need for rapid threat analysis can be 
urgent. Unfortunately, without some human intelligence or other 
highly classified source data, input to any threat analysis is 
constrained to dimensional information gained from photographs of 
the eguipment's antennas. In the past, intelligence agencies 
analyzing each new system on a case by case basis found this 
process to be very slow, tedious, and man power intensive. 

With the advent of powerful personal computers and the 
availability of sophisticated mathematics software, antenna 
analysis using data obtained from photographic intelligence may 
now be achieved in a rapid manner. The goal of this report is to 
document the software developed to accomplish this type of 
performance appraisal. With this report and its programs, 
NAVMARINTCEN is supplied with a user friendly tool to aid in 
their task of antenna system evaluation. 

It should be recognized that computer analysis of antenna 
parameters has its limitations. For example, it is impossible to 
account for the effects of adjacent structures on far-field 
radiation patterns and still keep the Mathcad applications 
moderately simple. In addition, without knowledge of parameters 
such as feed line characteristic impedance and antenna materials, 

2 



it is impossible to precisely assess the efficiency, gain, and 
radiated power of any antenna. Nevertheless, in most cases this 
report should provide the tools necessary to gain an excellent 
initial insight into the capabilities of systems which use the 
antenna types considered herein. 

Each chapter of this report reviews a specific type of 
antenna and is written as a comprehensive reference for the 
software. Copies of each application are included as appendices 
to provide the user with a printed illustration of the software. 



III. THE HELICAL ANTENNA 

The helical antenna is a wideband, highly directional device 
when operated in the axial radiation mode. It is commonly used 
in satellite and communications systems. To implement the 
Mathcad software for the helical antenna the following dimensions 
are needed: 

D = diameter of helix (center to center of the conductor) 

S = turn spacing (center to center of the conductor) 

L = length of one turn 

n = number of turns 

d = diameter of helix conductor 

C = circumference of the helix = 7rD* 

a = pitch angle = tan" 1 (S/7TD) " 
The first four dimensions are application inputs, the indicates 
that the remaining two parameters are calculated by the helical 
antenna Mathcad application. The helical antenna geometric 
relationships are illustrated in Figures 3.1 and 3.2. 



s 




L 



Distance along 
conductor between 
arrows 



FIGURE 3.1 Helix Dimensions 





s 


c 


-v L 



FIGURE 3.2 Pitch Angle of a Helix 

The axial mode helical antenna has a highly directive main 
lobe, negligible mutual impedance with adjacent antennas, a low 
voltage standing wave ratio (VSWR) , and a resistive input 
impedance if the following are conditions met [Ref 2: pp. 277- 
288] : 



B<,C x zl.l5 (wavelengths) 



(3.1) 



n>2 (turns) 



(3.2) 



12°^a^l4° (degrees) 



(3.3) 



Subscripts containing (A) indicate the dimension in wavelengths. 

Assuming (3.1) - (3.3) are satisfied, one can estimate 
directivity (D ) as follows: 

Do~i2C x 2 nS x (diwensionless) (3.4) 

For a long helix (nS A > 1) , the relative phase velocity of 
the traveling wave (p) is the key variable for calculating far- 
field radiation patterns and associated parameters. Although 
several equations can be used for determining relative phase 
velocity, the one which most closely matches measured results is 
[Ref 2: pp. 288-300] : 

L, 

p= r— } r (dimensionless) (3.5) 

S k +w+ (l/2n) 

In (3.5), (m) corresponds to the transmission mode number of the 
antenna. The transmission mode is a term used to describe the 
manner in which an electromagnetic wave propagates down the 
helix. The number assigned to a given transmission mode (T m ) is 
an integer. When m = the helix radiates in what is termed the 
normal mode, since the main lobe is perpendicular to the axis of 
the helix. In some texts a helical antenna which radiates in the 
normal mode is called an electrically small antenna. The normal 
mode is not commonly used and will not be covered further in this 
report. The Mathcad applications analyze only the non-zero 
transmission modes of a given helical antenna. 

The mode of a helix is determined by its physical size, with 
higher modes corresponding to larger antennas. The relationship 



between helix circumference and spacing for m = 1,2 is 
illustrated in Figure 3.3. 



2.6 



2.4- 



£ 2.2 

U> 

C 

I 2 





E 

3 


.!= 1.4F 



1.2 









































































Tl = 2 ' 




















fT= 1 


















































/ 





















0.2 0.4 0.6 0. 



Spacing in Wavelengths 



1.2 1.4 1.6 1.8 2 



FIGURE 3.3 Helix Mode Chart 

The general relationship between helical radiation mode, 
turn length, circumference, and spacing is provided in the 
following equations [Ref 2: p. 289]: 



L 2 =C 2 +S 2 



(3.7) 



^±=S k +m (3.6) 

P 

Once the relative phase velocity has been determined and a 
transmission mode is selected, it is possible to resolve the far- 
field radiation pattern of the helix. As long as the helix is 
long, it can be regarded as an array consisting of (n) one turn 
loops. To begin radiation pattern computations the phase shift 
(lj;) of each equivalent point source in the effective array factor 
of the helix is computed as follows: 

\lr = 27i (5,cos6-— ) (radians) (3.8) 

P 

In (3.8), (0) corresponds to the coaltitude, or deflection angle 

from the axis of the helix. 

As a result of the symmetrical nature of a helical antenna's 

main lobe the following relation holds: 

E^jE^ (V/m) (3.9) 

The far-field radiation pattern of a single helical turn is 
reasonably estimated by cos (8). The electric field pattern (E) 
is given by the product of the array factor and the individual 
turn's pattern. As predicted by the principle of pattern 
multiplication, the array factor corresponding to an array of 
isotropic point sources dominates the field pattern generated by 
a single turn of the helix. This effect can be seen in the 
following formula for electric field [Ref 2: pp. 294-295]: 

It should be noted that unless the helix is very short (nS A 



sin^* 

£-=sin — ( — )cos0 (V/m) (3.10) 

212 sin* 
2 

< .5), ground plane reflections and their effects on electric 

field patterns for the antenna are negligible. Consequently, 

ground parameters are not required for this application. 

The radiation intensity (U) at any far-field observation 

point is a function of E e and E per the following equation [Ref 

3 : pp. 28-29] : 

U=-±- [\E Q \ 2 + \EJ 2 ] (W/ solid ang) (3.11) 

2T) 

From (3.9), (3.11) can be reduced to: 

U=— \E\ 2 (W/ solid ang) (3.12) 

no 

In (3.11) and (3.12), (77J is the intrinsic impedance of free 
space. 

The average radiated power (P rad ) for any antenna is given 
by: 

p rad=[f UdQ=f 2n rUsinedQd<b (W) (3.13) 

In (3.13), (n) is a sphere in the far-field surrounding the 
antenna. 

It is impossible to determine total efficiency (e t ) of the 
helical antenna based only on dimensional information. Therefore, 
an antenna's gain (G) cannot be precisely determined using the 
following general gain formula: 

10 



G=e c D (dimensionless) (3.14) 

However, a unique feature of helical antennas is that input 
impedance (ZJ is essentially equal to input resistance (R) when 
(3.1) - (3.3) are satisfied. Fortuitously, the input resistance 
of the helical antenna can be calculated with observed 
measurements by [Ref 2: pp 277-278]: 

Axial Feed: R=l4 OC, (fi) (3.15) 



Peripheral Feed: i?=-i^ (Q) (3.16) 



If antenna feed characteristic impedance (Z ) is known or 
can be estimated, then reflection efficiency (e r ) can be computed 
from the voltage reflection coefficient (T) by [Ref 4: p. 460]: 

T= (dimensionless) (3.17) 

R+Z n 

e r =l-|r| 2 (dimensionless) (3.18) 

Although (3.17) and (3.18) provide an estimate of reflection 
efficiency, no other helical antenna efficiency terms can be 
determined based on geometry alone. Thus, all other components 
of total efficiency are assumed to be unity and gain is expressed 
as: 

G=e r D (dimensionless) (3.19) 

EIRP is a commonly used term from communications that is 
formally defined as the product of antenna gain and total power 

11 



accepted by the antenna from the transmitter. EIRP is determined 
as follows [Ref 5: p. 62]: 

EIRP=P iad D Q (W) (3.20) 

A functional helical antenna will exhibit nearly circular 
polarization when (3.1) - (3.3) are satisfied. Axial ratio (AR) 
provides a figure of merit for circular polarization in that if 
it is equal to unity the polarization of the antenna is exactly 
circular. The further axial ratio is from one, the more 
elliptically polarized the helical wave will be . The axial ratio 
of a helix is [Ref 2: pp. 301-307]: 

AR= \L X (sin (a) -l/p) | (dimensionless) (3.21) 

The helical antenna's unit polarization vector (a a ) at a 
given point in the far-field is computed using the Cartesian 
components of electric field. The Cartesian components of 
electric field and the antenna's unit polarization vector are 
determined using the results of (3.9) and (3.10) as follows [Ref 
5: p. 555] : 

E x =E e cos (0) cos (<|>) -£ , 4) sin(4)) (V/m) (3.22) 

E y =E 6 cos(d) sin (4>) +£^003(4)) (V/m) (3.23) 

E z =-E e sin(d) (V/m) (3.24) 

aj? +aJZ +a E 
o a (x,y, z) - — ^— ^ — - (dimensionless) (3.25) 

y/\E(x,y,z) | 2 
In (3.25), (a xyz ) are the Cartesian unit vectors. 

12 



When the helical antenna is used for reception and the 
incoming wave's electric field unit vector (p w ) at a given point 
in the far-field is known or can be estimated, the polarization 
loss factor (PLF) is given by [Ref 3: p. 51]: 

PLF=|o^o*| 2 (dimensionless) (3.26) 

The term which best describes an antenna's ability to 
capture incoming electromagnetic waves and extract power from 
them is maximum effective aperture (A em ) . Maximum effective 
aperture for a helical antenna is [Ref 3: p. 63]: 

A ew -[PLF][e z ^-D } (m 2 ) (3.27) 

471 

In (3.27), (A) is wavelength of the frequency (f) of interest. 
Although the current (I ) at the terminals of the helical 
antenna cannot be determined from dimensional information alone, 
if the current is assumed to be unity the radiation resistance 
(R r ) and maximum effective height (h em ) are estimated by [Ref 2: 
p. 42]: 

^ = 4-^77 <G) (3.28) 

\lo\ 2 



h errr 2 ' 






Helical antenna Mathcad applications are valid for conductor 
diameters given by: 

.005A.scte.05A. (m) (3.30) 

13 



The bandwidth (BW) of an operational helix is determined by 
the high and low frequencies (f high/ fi ow ) corresponding to the 
dimensional limits of (3.1). Therefore, bandwidth can be 
calculated using the speed of light (c) by: 



1.15 c 

C 



ftlsfiT ± ^^ <**> ( 3 - 31 ) 



BW=f hlgh -f low (Hz) (3.33) 

Most of the parameters calculated by the Mathcad helical 
antenna applications are only valid if the observation point (r) 
is in the far-field. An observation point is considered to be in 
the far-field if all of the following are satisfied [Ref 3: p. 
92] : 

rkl.6\ (m) (3.34) 

r*5nS (m) (3.35) 

r± 2 ^ 5 * 2 (m) (3.36) 

A 

Table 3.1 and Figure 3.4 compare measured data to that 
calculated by the Mathcad applications for a 10 turn helical 
antenna (D = .1074 meters and a = 12.8°) [Ref 6: 13-6 - 13-9]. 



14 



TABLE 3.1 Helical Antenna Data Comparison 



ANTENNA 
PARAMETER 


MEASURED 
DATA 


CALCULATED 
DATA 


HALF -POWER 
BEAMWIDTH 


39° 


37° 


GAIN 


12.5 dB 


14.3 dB 



15 



Comparison of Helical Antenna Electric Fields 



> 



■o 
o 



2 



o 

l±j 



1 
0.8 
0.6 

0.4 

0.2 



•0.2 

■0.4 

■0,6 

•o.s 



— Calculated Electric F 
Measured Electric Fie lid 




feld 






FIGURE 3.4 Helical Antenna Electric Field Pattern 



16 



IV. THE BEVERAGE ANTENNA 

The Beverage antenna is a single wire antenna parallel to 
the ground and terminated with a load equal to the characteristic 
impedance (Z ) of the wire. The transmitter or receiver of a 
Beverage antenna has one end connected to the wire and the other 
to ground. Because of its matched termination, the Beverage 
antenna does not develop a significant standing voltage wave 
along its length. Therefore, it is known as a traveling wave 
antenna. The relative phase velocity (p) of the wave traveling 
down the antenna is typically less than one. Thus, the Beverage 
antenna is also considered a slow wave antenna. [Ref 3: pp. 372- 
374] 

Although radiation can occur at any non-uniformities in the 
device, the Beverage antenna primarily generates a vertically 
polarized cone shaped main beam that points in the direction of 
the traveling wave. The geometry of a Beverage antenna is 
illustrated in Figures 4.1 and 4.2. 



17 



L 

4 fc 


^ W 





h Zo? 


Ground 



FIGURE 4 . 1 The Beverage Antenna 



18 




FIGURE 4.2 Elevation Angle (6) of a Beverage Antenna 



19 



In Figure 4.1, (h) is the antenna's height above ground and (L) 
is the total length of the antenna. In Figure 4.2, (6) is the 
angle of incidence of an incoming or transmitted wave with 
respect to ground. 

Typically, the electrical length of a Beverage antenna (L A ) 
will be on the order of 0.5 to 2 wavelengths. Maximum length 
( L Amax) a "t which the antenna is expected to operate is a function 
of both arrival angle of the incoming wave and relative phase 
velocity. A precise formula for maximum length is [Ref 7: p. 14]: 

L xmax = : (wavelengths) 

4(--cos(6)) { ' 

P 

Unfortunately, use of (4.1) is normally not possible. The 

wave's angle of incidence is always changing and can only be 

estimated using statistical techniques. In addition, relative 

phase velocity is not easily determined by the antenna's geometry 

and, consequently is not generally known. Some relative phase 

velocity measurements have been conducted over the following 

frequency band [Ref 7: p. 19]: 

1.6 MHz < f < 10.5 MHz (Hz) (4.2) 

If the frequency of interest meets the criteria of (4.2), then 
(p) can be computed by: 

p= .65891(— ^) -"8523821 ( dimensionless) (4 * 3) 

1000 

If inadequate information is available to use (4.1), the 
Beverage antenna application user can estimate maximum length 

20 



from Table 4.1 [Ref 7: p. 14]. 

TABLE 4.1 Maximum Effective Length of a Beverage 

Antenna in Wavelengths 



6 in 
deg 


■ Lj Amax 


p=.89 


p=.91 


p=.93 





2.02 


2.53 


3.32 


10 


1.80 


2.19 


2.76 


20 


1.36 


1.57 


1.84 


30 


.97 


1.07 


1.19 


40 


.70 


.75 


.81 


50 


.52 


.55 


.58 



Because of the difficulty determining (6) and (p) , the 
Mathcad application assumes that the difference between 
frequencies corresponding to 0.5 to 2 wavelengths is the 
bandwidth for the Beverage antenna. The application also 
computes relative phase velocity per (4.3), but the user is 
cautioned that the frequencies of interest must satisfy (4.2). 

The Beverage antenna transmits or receives vertically 
polarized waves. In the case of reception, the question might 
arise as to how a wire lying parallel to the ground can receive a 
vertically polarized signal. For higher frequency operations 
that utilize sky wave propagation, the tilt of the incoming wave 
provides a horizontal component of the vertically polarized 
electric field (E) with respect to the ground and the antenna. 
It is the horizontal component of the wave that is parallel to 



21 



the antenna which generates the emf on the wire. 

For lower frequencies (i.e., < 300 Khz), the physics of a 
Beverage antenna is much more complex. In lower frequency 
applications, the ground wave is the principle propagation path. 
In this situation there is negligible tilt to the wave as a 
result of propagation path geometry. However, as the vertically 
polarized wave travels over an imperfect conductor the electric 
field closest to ground begins to develop a forward tilt as 
pictured in Figure 4.3. As in the case of the higher frequency 
applications, the tilted electric field of the low frequency wave 
has a horizontal component parallel to the antenna which induces 
an emf on the wire. 

Originally, use of the Beverage antenna was restricted to 
low frequencies propagating over very poor ground. The Beverage 
antenna is now often used in an attempt to reduce noise 
interference in high frequency operations over excellent ground. 
The higher frequency skywave propagation path provides the 
necessary tilt to receive vertically polarized signals, but the 
near perfect ground does tilt vertically polarized ground waves 
from nearby noise sources. Thus, the high frequency Beverage 
antenna becomes a highly directive, low noise device. 



22 



i 


tLU 




E x <^ 


^r ^^^^ 


Ground 



FIGURE 4.3 E Field Over an Imperfect Ground 



23 



Development of the electric field pattern of a Beverage 
antenna begins with an understanding of the current on the 
antenna. If one assumes low ohmic losses, matched termination, 
and negligible attenuation along the wire, the phasor current 
amplitude is constant and the phase velocity is that of free 
space; hence [Ref 5: p. 240]: 

I(z)=I e- jkz (A) (4.4) 

In (4.4) the antenna is assumed to lie along the +z axis, (I ) is 
current at the transmitter's terminals, and (k) is the free space 
wavenumber given by: 

k=ll_ (in -i) (4.5) 

In (4.5), (X) is wavelength of the frequency (f) of interest. 
Beverage antenna Mathcad applications assume (I ) is normalized 
to one amp. 

With the current defined by (4.4) the magnitude of the 
Beverage antenna's electric field is obtained by [Ref 8: pp. 315- 
316] : 

30^J sin(6> si n (AT) , , (4 . 6) 

I X 

In (4.6), (r) is the distance from the antenna to the far-field 
observation point, L is the length of the antenna, and (X) is 
given by: 



24 



X=— (l-cos(B)) (radians) (4.7) 

The electric field pattern given by (4.6) is rotated about 
the +z axis to form the three-dimensional field pattern above the 
ground plane. The pattern is only valid, however, in the far- 
field. Therefore, all of the following conditions must be 
satisfied for (4.6) to apply [Ref 3: p. 92]: 

rzl.6\ (m) (4.8) 

r;>5L (m) (4.9) 

r;>-^ (m) (4.10) 

Improvements to the accuracy of (4.6) can be made for far- 
field radiation patterns if one accounts for the effects of real 
ground. Through use of image theory and the fact that Beverage 
antennas excite vertically polarized waves, the electric field 
pattern equation is modified as follows [Ref 5: pp. 229-235]: 

30icLJ sin(e) Bln(J0 -j^coscf -e) (4>11J 

I ' X ' l v 

In (4.11), (r v ) is the vertical reflection coefficient of 
ground. Antenna height is typically less than one wavelength. 
The vertical reflection coefficient is given by: 
In (4.12), (e r .) is the relative complex permittivity of the 
ground under the antenna and is calculated as follows: 
In (4.13), (e r ) is the relative permittivity of the ground and 



25 



r = 



e r ,cos ( -^ -6) -* e r /-sin s ( - -6) 
1 2 \ r 2 



e r /cos ( — -6) +a 

r 2 \ 



{dimensionless) (4.12) 



e ,-sin 2 (--6) 
2 



e r /=e_-j ■=— {dimensionless) (4.13) 

1 r 2nfe 

(a) is the conductivity of the ground. 

The direction of maximum radiation of a Beverage antenna may 
be determined from (4.11). However, it can also be estimated 
guickly by the following empirical formula [Ref 5: p. 241]: 

Wcos-Ml-^P) (radians) (4>14) 

"X 

With the magnitude of the vertically polarized electric 
field given by (4.11), radiation intensity is computed as follows 
[Ref 3 : pp. 28] : 

U=^^[\E\ 2 ] (w/ solid ang) (4.15) 

2T lo 

In (4.15), (rj ) is the intrinsic impedance of free space. 

The radiated power and directivity of a Beverage antenna are 
determined by applying radiation intensity to standard antenna 
formulas as follows [Ref 3: pp. 28-30]: 



26 



P rad = Tf Vsin(0)d6d4> (W) (4.16) 

Jo Jo 

D Q = 4llUi max) (dimensionless) (4.17) 

The characteristic impedance of a Beverage antenna can 
be estimated by its dimensions and is generally resistive. 
Characteristic impedance of a Beverage antenna over perfect 
ground is given by [Ref 7: pp 19-21]: 



h 
d 



Z =1381og(4^) (Q) (4.18) 



In (4.18) , (d) is the diameter of the wire in the same units as 
(h) . Caution must be exercised when using this value of 
characteristic impedance since any sharp transition in the wire 
(i.e., vertical downleads) or real ground effects can reduce the 
accuracy of the calculation. Typical values for a Beverage 
antenna's characteristic impedance are 200-300 ohms. 

If the impedance (Z x ) of the terminating load of a Beverage 
antenna is known or can be estimated, reflection efficiency (e r ) 
can be determined from the voltage reflection coefficient (T) as 
follows: 

r=^C£° (dimensionless) (4 ' 19) 

z 1+ z 

e r =l-|T| 2 (dimensionless) (4.20) 

Other than reflection efficiency, accurate estimates of 



27 



other Beverage antenna losses cannot be determined by geometry 
alone. Nevertheless, there are other sources of lost power. 
Since a Beverage antenna is a relatively long antenna with a 
matched termination, relatively little power is reflected by the 
load. Instead, most of the power supplied by the transmitter 
that is not radiated is absorbed by the load or lost as heat to 
the ground. 

Gain (G) is the product of antenna's directivity and 
efficiencies. The Mathcad Beverage antenna applications express 
gain as [Ref 3: p. 43]: 

G=e z D (dimensionless) (4.21) 

Effective isotropic radiated power for the Beverage antenna 
(EIRP) is the product of the power radiated by the antenna and 
the directivity. (EIRP) is computed by [Ref 5: p. 62]: 

EIRP=P iad D (W) (4.22) 

Electromagnetic waves incident upon a Beverage antenna are 
normally assumed to be vertically polarized. If the incoming 
wave is not vertically polarized, a polarization mismatch 
occurs with the antenna and losses result. Polarization losses 
are determined from a polarization loss factor (PLF) given as: 

PLF=\o w -o* a \ 2 (dimensionless) (4.23) 

In (4.23), (a a ) and (a w ) are the unit polarization vectors of the 
antenna and wave, respectively. 

Maximum effective aperture (A em ) is estimated from 



28 



directivity, (PLF) , and reflection efficiency as follows [Ref 3: 
pp. 51-63]: 

A em =[PLF] [e z (-^)D Q ] (m 2 ) (4.24) 

The maximum effective height (h em ) of a Beverage antenna can 
be determined using the results of (4.24) as follows [Ref 5: p. 
42] : 



h=2 t 



R r A em , s (4.25) 

In (4.25), (R r ) is radiation resistance and is written as: 

R=^m (Q) (4.26) 

I T 2 
I - L o\ 

The conductor diameter must be much less than the length of 
the Beverage antenna to avoid unwanted radiation from vertical 
sections. For the purpose of Beverage antenna Mathcad 
applications, the following is assumed for proper antenna 
operations: 

d<.L if d<.01L (/77) (4.27) 

Table 4.2 and Figure 4.4 compare measured data to that 
calculated by the Mathcad applications for a Beverage antenna 
(L=110.4 meters and h=1.23 meters) operating over dry soil 
(a=.003 S/m and e r =12) at 18 MHz. Table 4.3 compares measured 
and calculated data for a Beverage antenna with L=110.4 meters 
and h=1.13 meters operating over wet soil (a=.01 S/m and e r =17) 



29 



at 5 MHz [Ref 9: pp. 22-26]. 



TABLE 4.2 Beverage Antenna Data Comparison 



ANTENNA 
PARAMETER 


MEASURED 
DATA 


CALCULATED 
DATA 


"max 


18.5° 


19.2° 



TABLE 4.3 Beverage Antenna Data Comparison 



ANTENNA 
PARAMETER 


MEASURED 
DATA 


CALCULATED 
DATA 


Zo 


450 n 


408 n 


P 


.93 


.91 



30 



Comparison of Beverage Antenna Electrio Fields 



> 

c 

"D 

'E 

D 

IZ 

o 






1 

0.8 

0.6 

0.4 

0.2 



-0.2 

-0.4 

-0.6 

-0.8 

-1 



ff » '» I '^ ".n.., 




CaJ^utejte'3' ''ElgctH'd^fi^fd-X.'So.ljd 'Uriel .. %< / 

''0easu'red.--€leciric / Rieid\ A Das-fed, Line 



"■■ i "'" 



FIGURE 4.4 Beverage Antenna Electric Field Patterns 



31 



V. THE LOOP ANTENNA 

A loop antenna is a coil of one or more turns. It is 
commonly used as a receiving antenna for operations in the lower 
frequency regions. The loop antenna is also used for direction 
finding and UHF transmissions [Ref 10: p. 6-1]. Loop antennas 
may have an air core or ferrite core. They may also be 
electrically large or small. For the purpose of the Mathcad 
applications, a loop antenna is considered electrically small if 
its radius (a) satisfies the following [Ref 3: p. 181]: 

a<-±- (m) (5.1) 

6 7T 

In (5.1), (A) is wavelength. The geometry of both large and 
small loop antennas is illustrated in Figure 5.1. 



32 




FIGURE 5.1 Loop Antenna Geometry 



33 



The radius of the conductor is (b) in Figure 5.1. For all loop 
antenna Mathcad applications the center of the loop is the origin 
and the antenna's axis is aligned parallel to the +z axis. When 
Mathcad applications examine the performance of a loop over a 
ground plane, the coordinate system is rotated with the antenna 
as necessary to obtain the desired geometry (i.e., the axis of a 
vertical loop is parallel to the ground and the +z axis) . 
A. THE ELECTRICALLY SMALL LOOP 

Electrically small loops are normally used for low freguency 
reception or direction finding. Small loops are poor 
transmitters due to small radiation resistance (R r ) and low 
conduction-dielectric efficiency (e cd ). Transmitter performance 
can be improved with increased perimeter, adding additional 
turns, or insertion of a ferrite core [Ref 3: p. 164], 

Two key assumptions are made in the analysis of a small 
loop. First, it is assumed that current around the loop is 
constant. This supposition allows the loop to be approximated by 
an infinitesimal magnetic dipole centered at the origin and 
parallel to the + z axis. Second, it is presumed that the various 
resistances and reactances of the loop can be computed from 
dimensional information and knowledge of the antenna's material 
properties. 

Given the above assumptions in free space, the electric 
field (E) for a small loop in the far-field is determined by [Ref 
3: pp. 168-169]: 
In (5.3), (S) is the cross-sectional area of the loop, (f) is the 

34 



E z =E e = (V/m) (5.2) 

JcSf^sinWe^ 
* 2r 

frequency of interest, (r) is the distance from the origin to the 
observation point in the far-field, (I ) is the antenna feed 
current, (/x ) is the permeability of free space, and (k) is the 
free space wavenumber given by: 

k= ^Y {w ' 1] (5 * 4) 

As is the case in all Mathcad applications, current in the 
loop is normalized to one amp. Since electric field is not a 
function of (0) , the field pattern is symmetric when rotated 
about the antenna's axis. 

It should be noted that (5.2) and (5.3) apply to all small 
loops, regardless of shape. Thus, (5.2) and (5.3) can be used 
for small square loops. However, it should also be noted that 
loop antenna Mathcad applications assume a circular loop is being 
analyzed and calculate cross-sectional area based on the radius 
provided by the user. In order to use the applications with 
rectangular loops or loops of an odd shape, the user must compute 
an equivalent radius (a) that will yield the correct area. 

An observation point for any loop antenna is assumed to be 
in the far-field if the following conditions are valid [Ref 3: 
p. 92] : 

r*1.6A (m) (5.5) 



35 



I^SD (m) (5.6) 

i±^- (in) (5.7) 

In (5.5)-(5.7), (D) is the largest dimension of the loop. The 
largest dimension of the loop is assumed to be the diameter. 

For small loops in free space, radiated power (P rad ) is 
estimated as follows: 

Pra^o(^) (^) 4 |JJ 2 (W) (5.8) 

In (5.8), (r) ) is the intrinsic impedance of free space. 

The directivity (D ) of a small loop in free space is 1.5 
and, ignoring polarization any mismatches, the maximum effective 
aperture (A em ) of a lossless loop is written as [Ref 3: p. 175]: 

The ohmic resistance of any loop antenna (R ohmic ) , including 
multiple turn antennas, is estimated by the following [Ref 3: 
pp. 171-172] : 

In (5.10), (N) is the number of turns, (R s ) is the surface 
impedance of the conductor, (R p ) is the ohmic resistance due to 
proximity effect, and (R ) is the ohmic skin effect resistance 
per unit length. If the conductivity (a c ) of the conductor is 
known, the surface impedance of the conductor is computed by: 

36 



R s = 



\ 



nf[i t 



(CI) 



(5.11) 



Given the spacing between turns (q) has been measured, the 
ratio of (R p ) to (R ) is estimated using Figure 5.2 [Ref 3: p. 
172] . 



1.5 



o 



\ 1 - 






0.5 



7< 



Number Adjacent to Line = N 




1.5 



2.5 3 3.5 

Spacing Ratio: q/2b 



FIGURE 5.2 R p /R 

The radiation resistance of a small loop in free space is 
determined using the circumference of the loop (C) as follows 
[Ref 3: pp. 170-171] : 



R =207T 2 (-f ) 4 N 2 (CI) 



(5.12) 



When the radiation and ohmic resistance of any antenna 



37 



has been calculated, the conduction-dielectric efficiency of the 
antenna is determined by: 

e cd = (dimemsionless) (5.13) 



R ohmic +R i 



From (5.10) and (5.12) it can be seen that ohmic resistance 
is directly proportional to the number of turns while radiation 
resistance is proportional to the square of the number of turns. 
Thus, as shown in (5.13) conduction-dielectric efficiency can be 
improved by increasing the number of turns in a loop antenna. It 
can also be seen in (5.10) and (5.12) that increasing the radius 
of the loop improves conduction-dielectric efficiency. 

Additional improvements to conduction-dielectric efficiency 
may be made by inserting a ferrite core in the loop antenna. If 
a core is added, (5.12) is modified as follows: 

r =20ti 2 (-£ ) 4 ( — ) 2 N 2 (Q) (5.14) 

Effective permeability of the ferrite core (n e ) in (5.14) is 
computed by: 

» e = n n —( 7T {H/m) (5 ' 15) 

In (5.15), (/i f ) is the actual permeability of the core material 
and (D demag ) is an experimentally derived demagnetization factor. 
Demagnetization factor as a function of the ratio of core length 
to diameter is shown in Figure 5.3 [Ref 3: pp. 196-197]. 



38 



10" 1 
o* 



E 

V 
V 

a 10 -2 

lT 


"D 


Li. 

C 
,0 

1 

1 10- 3 

O* 


E 
a 

Q 

10" 4 
10 














:::::::::::::::!:::::::::::*;:::: 










\. 


' i > 


• • • « 


























III 1 1 


























































D 




10 1 
Core Length/Diameter 


10 2 



FIGURE 5.3 Demagnetization Factor 



Any antenna parameter that requires use of permeability in 
its formula must be approximated in the Mathcad applications by 
replacing (jli ) with (/x e ) [Ref 11: pp. 86-89]. 

When, a loop antenna is actually employed, it is not in free 
space and real ground must be considered. Although real ground 
does not change the components of electric field given by (5.2), 
it does modify (E^) in (5.3). Since the orientation of the loop 
with respect to ground determines the polarization of the loop's 
electromagnetic wave, alignment of the antenna must be known 



39 



before electric field can be correctly computed. In order to 
keep the loop antenna Mathcad applications moderately simple, 
only horizontal and vertical loops are considered. 

Although modeled by an infinitesimal vertical magnetic 
dipole, a small horizontal loop (i.e., +z axis perpendicular to 
ground) has horizontally polarized electromagnetic waves. The 
horizontal reflection coefficient (T h ) is [Ref 5: pp. 229-230]: 



cos(e)- v /e r /-sin(6) 2 . 
F H = i- - (dimensionless) (5.16) 

cos (6) + v /e r /-sin(0) 2 

In (5.16), the relative complex permittivity (e r .) of the ground 
is calculated using the relative permittivity (e r ) and 
conductivity (a) of the ground by: 

e r /=e -j — ^-r— (dimensionless) (5.17) 

In (5.17), (e ) is the relative permittivity of free space. 

The total electric field of a small horizontal loop over 
ground is the sum of the direct path signal and ground reflected 
signal. One can use image theory to estimate the contribution to 
the far-field pattern by ground reflections. The total electric 
field expression for a small horizontal loop positioned a 
distance (h) above the ground is: 



kS f \x I s in (Q) 
2r~ 



= «ij» 2 i^uw fl . Jh _ re -j2khcos(6) ] {v/rn) (5.18) 

™hoi Or n 



Note in (5.18) that the image antenna is a vertical infinitesimal 
magnetic dipole. Thus, the contribution of the image is 



40 



subtracted from the contribution of the actual antenna. The 
Mathcad loop antenna applications ignore the minor contribution 
of the surface wave to the electric field. 

The total electric field of a vertical loop over real ground 
can be determined in a manner similar to (5.18), but the vertical 
reflection coefficient (T v ) must be used for the image antenna's 
contribution to the far-field pattern. The equation for vertical 
reflection coefficient is [Ref 5: pp. 231-232]: 



e r /cos ( — -0) -a 

r 2 \ 



e r /cos( — -6) 



e r ,-sin 2 ( — -8) 

2 

- {dimensionless) (5.19) 



, e /-sin 2 (--6) 
\ r 2 



For a vertically mounted small loop, the image antenna's 
contribution is added to that of the actual antenna. Thus, total 
electric field for a small vertical loop is written as: 

= icgr> J sin(e) e . jkr[1+T j2khcos(Q)] {v/m) (5#20) 

It should be noted by application users that (5.20) applies to 
loops located in a coordinate system that has been rotated with 
the axis of the loop such that the +z axis is parallel to the 
loop axis and the ground. 

If the electric field of a small loop antenna over real 
ground is known, several parameters can be computed using general 
antenna formulas. Far-field equations for radiation intensity 
(U) , radiated power, directivity, and radiation resistance are as 
follows: 



41 



U=^-\E\ 2 (W/ solid ang) (5.21) 

2T lo 



P rad={( Usin(Q)dQd<b (W) (5.22) 



D =4tc^^ (dimensionless) (5.23) 

Pzad 



l^ol 2 

In (5.22), (£7) is a half sphere that encloses a loop over ground 
in the far-field. In (5.23), (U max ) is the maximum radiation 
intensity anywhere on the half sphere as determined by applying 
(5.22) . 

A unique feature of small loop antennas is that most 
efficiency terms associated with it can be calculated from the 
loop's measurements. One of the reasons for this attribute is 
the fact that input reactance (XJ can be reasonably estimated 
from [Ref 5: pp. 102-103]: 

*,=2nfau n [ln(8-) -1.75] (Q) (5.25) 

O ft 

Input impedance (ZJ for the small loop is found using the 
formula: 

Z i =R i+ jX i (Q) (5.26) 

In (5.26), input resistance (RJ is the sum of radiation and 
ohmic resistance and is written as: 



42 



Ri=Rr+R oh mic (0) ( 5 - 27 ) 

If the characteristic impedance of the antenna's feed line 
is known or can be presumed, the voltage reflection coefficient 
(T) and reflection efficiency (e r ) are determined by: 

r= r ° (dimensionless) (5.28) 

G r =l-|r| 2 (dimensionless) (5.29) 

As previously discussed, the polarization of a small loop 
matches the orientation of the loop (i.e., a horizontal loop is 
horizontally polarized) . Thus, if the polarization of an 
incoming wave is known or can be estimated, one can use the dot 
product of the unit polarization vector of a small loop antenna 
(a a ) and the unit polarization vector of an incoming wave (a w ) to 
compute the polarization loss factor (PLF) as follows [Ref 3: p. 
51] : 

PLF=\a w 'd* a \ 2 (dimensionless) (5.30) 

With the efficiency and loss terms of the small loop 
estimated by (5.13), (5.29), and (5.30); gain (G) , maximum 
effective aperture (A em ) , and effective isotropic radiated power 
(EIRP) can be expressed as [Ref 3 pp. 43-63]: 



43 



G=e r e cd D (dimensionless) (5.31) 

A em =e z e cd p o ( PLF) ( |1 ) (/n 2 ) (5.32) 

EIRP=P iad D (W) (5.33) 

Maximum effective height (h em ) can be determined from the 
maximum effective aperture and radiation resistance of the 
antenna as follows [Ref 2: p. 42]: 



h en> =2, 



*A. {m) (5.34) 

N ^ 



Mathcad applications are developed assuming bandwidth (BW) 
of an antenna is the range of frequencies over which all 
computations will be valid. For the small loop antenna Mathcad 
applications to be valid, (5.1) must hold. Thus, the radius of 
the loop will define lowest operating frequency (f min ) , the 
highest operating frequency (f max ), and the bandwidth of the 
antenna as shown below: 

f min ^0 (Hz) (5.35) 



f ™*Th {Hz) (5 * 36) 



BW=f mx -f nin (HZ) (5.37) 

The electric field patterns and antenna parameters obtained 
using the small loop Mathcad applications for a small loop in 
free space are identical to published results obtained using 



44 



method of moments techniques [Ref 3: pp. 169-180]. A comparison 
of the results of the Mathcad applications with measured data for 
a small loop (a=.25 meters, b=.005 meters) located 2.5 meters 
above a reflecting plane (cr=6xl0 7 S/m, e r =l) receiving a 30 MHz 
signal is provided in Table 5.1 [Ref 6: p. 5-14]. 

TABLE 5.1 Small Loop Antenna Data Comparison 



ANTENNA 
PARAMETER 


MEASURED 
DATA 


CALCULATED 
DATA 


DIRECTIVITY 


7.0 dB 


6.5 dB 



B. THE ELECTRICALLY LARGE LOOP 

Electrically large loops are those loops that do not satisfy 
(5.1). Use of these antennas is somewhat rare, with radii 
exceeding one wavelength normally not practical. Large loops 
are, in general, significantly more difficult to analyze than 
small loops since large loops cannot be approximated by 
infinitesimal magnetic dipoles. In addition, the input reactance 
of a large loop antenna cannot be calculated from the loop's 
geometry. 

The polarization of a large horizontal loop over real ground 
is horizontal, but the polarization of a large vertical loop is 
not vertical. Due to the complexities of computing the 
polarization and ground reflection coefficients of a large 



45 



vertical loop, the Mathcad large loop antenna applications will 
only examine the free space and horizontal cases. 

To begin analysis of the far-field radiation pattern of a 
large loop in free space, one must assume that the current on the 
loop is constant. As illustrated in Figure 5.4, this is an 
approximation that deteriorates with the size of the loop [Ref 3: 
p. 184]. 



< 

E 



"0 

3 

Q. 

E 
< 

c 

(D 

L 
1_ 
D 

u 



y 












|ca=.1 






8 
7 
6 
5 














































































1 


ca=.2 






4 
3 




















1 


<a= 3 




























ka=.4 






2 
1 
























i 
n 





















20 



40 



60 



100 



20 



40 



60 



Angle from Loop Feed Point 



FIGURE 5.4 Current Magnitude Distribution for a Large Loop 

If one assumes that loop current is constant, (5.2) applies 
to large loop antennas. The other component of electric field 



46 



(E^,) is approximated by [Ref 3: pp. 176-178]: 

V ^y^ CtolnW) (5-38, 

In (5.38), J x (kasin(0)) is a Bessel function of the first kind of 
order one. With the horizontal reflection coefficient of the 
ground computed per (5.16), total electric field of the large 
horizontal loop over real ground is given by: 

E h0 r= ^^^ e-^J.ikasinm ) {539) 

* [l-T h e^ 2khcos{e) ] (V/m) 

Radiated power for a large loop in free space is computed 
using (5.22), where radiation intensity is given by (5.21) and 
electric field is computed by (5.38). It should be noted that 
the far-field conditions of (5.5) -(5.7) must be satisfied if the 
Mathcad large loop antenna electric field and radiated power 
calculations are to be valid. 

Large loop, free space approximations for radiation 
resistance, directivity, maximum effective aperture, and maximum 
radiation intensity are as follows [Ref 3: p. 181] [Ref 11: pp. 
78-79] : 



47 



R i =60k 2 (-^)N 2 (CI) (5.40) 

A 



D =.682(-^) (dimensionless) (5.41) 

A 



A em =.0543UO (/?? 2 ) (5.42) 

g max = <2icfa|l )' ! |j | 2 (|584) 2 (F // so 2id ang ) (5.43) 

8T lo 

For the horizontal loop over real ground, radiation 
intensity is given by (5.21), where the electric field is 
computed in (5.39). With radiation intensity known, radiated 
power is calculated by (5.22) and effective isotropic radiated 
power is computed using (5.33). Radiation resistance and 
directivity for the horizontal loop are determined using generic 
antenna formulas as follows: 



I J, 



R z = ^ (CI) (5.44) 



■ o\ 



47t[7 „ 
D max (dimensionlesss) (5.45) 

Op 

r z&d 

Ohmic resistance for the large loop antenna is given by 
(5.8). With radiation resistance given by (5.40) or (5.44), as 
applicable, the conduction-dielectric efficiency of the antenna 
is computed using (5.13). 

Without knowledge of the input reactance, it is impossible 
to compute reflection efficiency of a large loop antenna. 



48 



Therefore, Mathcad applications for large loops assume a factor 
of unity for reflection efficiency and calculate gain as follows: 

G=e cd D Q (dimensionless) (5.46) 

In general, the polarization loss factor given in (5.30) 
cannot be determined for an arbitrary large loop. Thus, (PLF) 
for large loops is assumed to be unity and maximum effective 
aperture is given by: 

With maximum effective aperture determined for a large loop 
using (5.47), maximum effective height is found by applying 
(5.34) . 

The bandwidth of a large loop is determined by the loop's 
radius as follows: 

— <<a<A (/7?) (5.48) 

6 71 



<f<- (Hz) (5.49) 



6571 



BW=—(1-—) (Hz) (5.50) 

a 6tz 

Figure 5.5 compares the electric field pattern obtained from 
the Mathcad applications for a large loop (a=.46 meters, b=.05 
meters) in free space receiving a 330 MHz signal with published 
results obtained from method of moments techniques [Ref 3: p. 
180]. Table 5.2 compares antenna parameters calculated using the 
Mathcad applications with measured results for a large loop 

49 



(a=.46 meters, b=.05 meters) located .72 meters above a 
reflecting ground plane (a=6xl0 7 S/m, e r =l) receiving a 104 MHz 
signal [Ref 6: p. 5-14]. 

TABLE 5.2 Large Loop Antenna Data Comparison 



ANTENNA 
PARAMETER 


MEASURED 
DATA 


CALCULATED 
DATA 


INPUT RESISTANCE 


150 n 


207 n 


DIRECTIVITY 


7.0 dB 


6.7 dB 



50 




FIGURE 5.5 Large Loop Antenna Electric Field Patterns 



51 



VI. THE BEDSPRING ANTENNA 

The vertical directivity (D ) of a horizontal dipole over 
real ground may be improved by placing identical elements in a 
straight line above the original. A vertical line of horizontal 
dipoles is commonly referred to as a bay. In a similar manner, 
one may place additional bays adjacent to the original and 
achieve an improvement in horizontal directivity. Further 
improvements to the gain (G) of this array of bays may be 
realized by placing a reflector on one side to simulate the 
existence of a conducting plane. The entire arrangement of bays 
and reflector is referred to as either a bedspring or a curtain 
antenna. A bedspring array is freguently used for high frequency 
(3-30 MHz) short wave radio systems [Ref 10: pp. 21-1 - 21-6]. A 
typical bedspring antenna arrangement is illustrated in Figure 
6. 1. 



52 



REFLECTOR - 




DIPOLES - 




FIGURE 6.1 A Typical Bedspring Antenna 



53 



To implement the Mathcad software for the bedspring antenna 
applications, the following dimensions are needed: 
Z L = height of the i th element above ground 
Z 1 - Z i . 1 = vertical spacing between the i th and the (i-l) th 

element 
Y 1 = horizontal position (dipole center) of the 

i th bay (Y = 0) 
Yi - Y L . X = horizontal spacing between the i th and the (i-l) th 

bay 
N = number of bays 

M = number of elements in each bay 
X x = reflector position 
1 = half-length of each element 
The geometry for the bedspring antenna is shown in Figure 6.2. 



54 



X 



7 



* 



REFLECTOR 




th 



i Tn Element 



► y 



FIGURE 6.2 Bedspring Antenna Geometry 



55 



In Figure 6.2 , (ty) is the angle between the +y axis and the 
vector (r) from the origin to the observation point in the far- 
field. Bedspring antenna Mathcad applications assume that all 
antenna elements are identical horizontal dipoles parallel to 
each other and the y axis. The applications also presume that 
the reflector is a perfect vertical conducting plane, parallel to 
the y-z plane, and located in the -x half-space. 

The assumption that the reflector is a perfect conducting 
plane is fairly reasonable, as experimental data has shown that 
the reflector typically improves gain by 2.5 to 3.0 dB. The 
reflector screen may be constructed from tuned elements, such as 
half-wave dipoles, or it may consist of a pattern of closely 
spaced, parallel wires. The reflector is normally located about 
one-quarter wavelength behind the antenna [Ref 10: p. 21-6]. 

The electric field (E) pattern for the bedspring antenna is 
computed using the principle of pattern multiplication for 
antenna arrays. Using the identity cos (\|j) =sin(0) cos (0) , one may 
predict the electric field components of a single bay. If all 
elements in the bay are excited by a sinusoidal current with 
maximum amplitude (I m ) , the electric field components are [Ref 
12: pp. 229-231]: 

E 6l = -j60I m ^^ [cos {klsln (6) sin ^ } ~ cos ikl) ] sin (4>) cos (6) [A] 
61 m r [l-sin 2 (e)sin 2 (<t>)] 

(V/m) 

(6.1) 



56 



Ejk =j60I ±^j [cos (klsin (6) sin (4» ) -cos (kl) ] CQS ( ^ ) [fi] 
** m r [l-sin 2 (6)sin 2 (4))] (6.2) 

(V/in) 

Given the wavelength (A) of the frequency of interest, the 
wavenumber (k) in (6.1) and (6.2) is given by: 

* = ir (/77l) (6 * 3) 

The complex coefficients (A) and (B) in (6.1) and (6.2), 
respectively, are computed as follows: 



^_y>« q e JkZ 1 (cos(Qj -cos(6 ) ) r-^-R Q-J 2kz i cos < e > ] 

(dimensionless) 



(6.4) 



(6.5) 



o \~>M /-, „2kZj (cos (6) -cos(6J ) r - J „ _ -j2kZiCos (6) ■■ 

( dimensi onl ess) 
The relative amplitude of excitation (CJ of the i th dipole with 
respect to the first element in the first bay is required for 
both (6.4) and (6.5). Equations (6.4) and (6.5) also use (0 O ) to 
represent the desired vertical scan angle of the antenna. The 
vertical scan angle is approximately equal to the progressive 
phase shift from one element to the next in the bay. In (6.4), 
(Rv) is the vertical reflection coefficient over real ground. In 
(6.5), (R h ) is the horizontal reflection coefficient over real 
ground. These reflection coefficients are determined by [Ref 5: 
pp. 229-235]: 



57 



€ r /cos(0) - v /e r /-sin 2 (6) . 

r v =— - v z (dimensionless) (6.6) 

e r /cos (6) + v /e r /-sin 2 (6) 
_ cos (0) - v /e r /-sin(0) 



2 

(dimensionless) (6.7) 



COS (0) +^£^.'-510 (0) 2 

The relative complex permittivity (e r .) of the ground needed 
to determine both reflection coefficients is calculated using the 
relative permittivity (e r ) of the ground, the conductivity of the 
ground (a), and the permittivity of free space (e ) as follows: 

e r /=e r -j — ^-r— (dimensionless) (6.8) 

1 z 2nfe 

The electric field components of (6.1) and (6.2) apply only 
in the far-field. Thus, all of the following conditions must 
hold if the computed electric fields are to be valid [Ref 3: pp. 
92-93] : 

r> 1.6 X (m) (6.9) 

r>5D (/7?) (6.10) 

In (6.10) and (6.11), (D) is the largest physical dimension in 
any direction of the antenna and is equal to the bedspring's 
diagonal length. 

If one assumes that the amplitude and phase of the feed 
current in corresponding elements in each bay is the same, the 
array factor (S y ) for N bays is written as [Ref 12: pp. 229-231]: 

58 



5 y = £L e jky »-* sinie) tsin(4»-sin(* )] ( di/ne/2sio ^ iess ) (6 .i2) 

In (6.12), (<p ) is the azimuthal scan angle of the antenna. The 
azimuthal scan angle is approximately equal to the progressive 
phase shift between bays in the bedspring antenna. 

The array factor accounting for the perfect image from the 
reflector (S x ) is: 

S x=1 _ e -^i sin(6 > cos W (dimensionless) (6.13) 

The total electric field components (E et , E^.J are computed 
by taking the product of all appropriate array factors and the 
pattern for a single bay as follows: 

E Qz =E ei S y S x (V/m) (6.14) 

£^=25^5^ (V/m) (6.15) 

The radiation intensity (U) of the bedspring antenna is 
determined from (E et ) and (E^J by [Ref 3: pp. 27-29]: 

U=-£- [|£ et | 2 +|£^ t | 2 ] (W/solid ang) (6.16) 

In (6.16), (r) Q ) is the intrinsic impedance of free space. 

Using the radiation intensity calculated in (6.16), one may 
calculate radiated power (P rad ) by computing the following 
integral over the quarter sphere (fi) through which 
electromagnetic energy from the antenna flows: 

P rad =ff Usin(e)dBd$ (W) (6.17) 



59 



Mathcad bedspring antenna applications assume that each 
element is excited by a feed current with the same maximum 
amplitude. Although this assumption may seem restrictive, it has 
been experimentally determined that maximum gain is obtained from 
a bedspring antenna if all radiator currents are of egual 
amplitude. Therefore, the following approximation for antenna 
feed current (I D ) holds [Ref 10: p. 21-17]: 

\l \=M-N-\l m \ (amps) (6.18) 

It should be noted that the bedspring antenna applications 
normalize all element excitation currents to one amp. 

Given the magnitude of the feed current for the entire 
antenna, radiation resistance (R r ) for the complete assembly is 
determined as follows: 

^"T^TT < Q > (6.19) 

Directivity is computed from radiation intensity and 
radiated power by: 

A-ttTJ 

D o = ss* (dimensionless) (6.20) 

In (6.20), (U max ) is the maximum value of radiation intensity 
anywhere on the quarter sphere encompassing the antenna's 
emissions . 

Gain is the product of total antenna efficiency (e t ) and 
directivity. Gain is normally expressed as: 



60 



G=e c D (diwensionless) (6.21) 

Unfortunately, the total efficiency of a bedspring antenna 
cannot be easily determined from its dimensions. For example, 
input impedance is a key parameter needed to calculate an 
antenna's reflection efficiency. However, input impedance has 
several components that are extremely difficult to determine. 
Self impedance of each element, mutual impedance between real 
elements in the array, mutual impedance between real elements and 
image elements from the reflector, and mutual impedance between 
real elements and imperfect images in the ground plane all 
contribute to input impedance. Because of the complexity of 
computations, input impedance of the bedspring antenna can only 
be estimated by an extensive method of moments algorithm. 

Other problems computing antenna efficiency include the 
possible existence of tuning devices in the feed lines; probable 
lack of sinusoidal current distributions for elements of 
arbitrary length; and inability to properly model the ground, 
reflector screen, and characteristic impedance of the feed 
assembly. Nevertheless, bedspring antenna gain can be adequately 
estimated despite the extensive calculations associated with 
precise modeling. Experience has shown that a properly tuned 
bedspring antenna operating within the band of expected 
frequencies exhibits very little loss. Consequently, gain is 
estimated in Mathcad bedspring applications as [Ref 10: p. 21-3]: 

G=10log(D o ) -2 (dB) (6.22) 



61 



Effective Isotropic Radiated Power (EIRP) is the product of 
radiated power and directivity and is computed by [Ref 5: p. 62]: 

EIRP=P Iad D (W) (6.23) 

The bedspring antenna's unit polarization vector (a a ) may be 
found by converting the electric field components of (6.14) and 
(6.15) to Cartesian coordinates as follows [Ref 4: pp. 35-36, 
364-367] : 

E x =E e cos(Q)cos($) -E^sinify) (V/m) (6.24) 

E y =E e COS (6) sin ((J)) +^cos (<J>) (V/m) (6.25) 

E z =-E Q sin(Q) {v/m) (6.26) 

_ a.£+a.E+a z E^ . 

o a (x, v, z) = — Z— * — - ( dimension! ess) (6.27) 

y/\E{x,y,z) | 2 

In (6.27), (a x y J are the unit vectors for the Cartesian 
coordinate system. 

The polarization loss factor (PLF) of a bedspring antenna at 
a point in the far-field for a given incoming wave with unit 
polarization vector (a w ) is expressed as [Ref 3: p. 51]; 

PLF=\o w -o* a \ 2 (dimensionless) (6.28) 

Without knowledge of antenna efficiencies, one cannot 
exactly predict an antenna's maximum effective aperture (A em ) or 
maximum effective height (h em ) . Nonetheless, one may assume a 
lossless antenna system and approximate these parameters as 
follows [Ref 2: pp. 29-43]: 

62 



A eir =D (PLF) <-£i) (m 2 ) (6.29) 



A.n-2. 



££»! (m) (6.30) 



For a half-wave dipole assembly, such as that pictured in 
Figure 6.1, a complex current that achieves maximum gain is fed 
to each element only within a few percent of the frequency (f A/2 ) 
whose wavelength matches the length of the half -wavelength 
dipoles. Therefore, the bandwidth (BW) of a Figure 6.1 type 
bedspring antenna is [Ref 10: p. 21-16]: 

f higb =1.02f x/2 (Hz) (6.31) 

^iou=-58f, /2 (Hz) (6.32) 

BW=f high -f Jow (HZ) (6.33) 

In (6.31), (f h igh) is the upper frequency of the antenna. In 
(6.32), (f low ) is the lower bound on operating frequency. 

If wideband operations are required for a bedspring antenna, 
a symmetrical feed arrangement as shown Figure 6.3 may be 
employed . 



63 



REFLECTOR- 



Dl POLE ASSEMBLY - 




- TRANSMITTER 



FIGURE 6.3 Symmetrical Feed Bedspring Antenna 



64 



The bandwidth for a symmetrical feed bedspring antenna is 
given by [Ref 10: p. 21-16]: 

£ high = 1.5£ x/2 (Hz) (6.34) 

f low =.98f x/2 (Hz) (6.35) 

BW=f high -f low (Hz) (6.36) 

Figure 6.4 compares the electric field pattern computed by 
the bedspring antenna Mathcad applications with measured results 
for a two-bay (Yi=26 meters) , four-stack (Z x =13 meters) bedspring 
antenna with reflector (X x =7 meters) operating at 10 MHz over 
soil (o=.01 S/m, e r =10) [Ref 12: p. 115]. 



65 



Comparison of Bedspring Antenno Electric Fields 



c 

> 






u 



o 

u 



0.8 
0.6 

0.4 
0.2 



n - 



-0.2- 



b -0.4- 



-0.6- 




•:::::::;iiifjc 



Cole ujjaitfd El'e ct;io""F"it!;d - ,;.S"o \ itMJ/ 1 e 

•Measured NElsc'tric/Fi eld/ 4 Dosfre.d' Dae 



-0.5 h 



-1 



FIGURE 6.4 Bedspring Antenna Electric Field Patterns 



66 



VII. THE SPIRAL ANTENNA 

Spiral antennas are a family of two- and three-dimensional 
structures that maintain a constant input impedance, beam 
pattern, gain, and polarization as well as many other parameters 
over a wide range of freguencies. Spiral antennas are commonly 
referred to as freguency independent, or broadband, devices. A 
two-dimensional spiral is called a planar spiral, while a three- 
dimensional spiral is usually termed a conical spiral. Planar 
and conical spiral antennas are commonly used in applications 
such as direction finding, missile guidance, and satellite 
tracking. [Ref 6: pp. 14-2 - 14-3] 

Although there are several types of planar and conical 
spiral antennas, Mathcad applications will fully analyze only 
those antennas with reasonably simple, closed form eguations: 
eguiangular planar spirals and conical log-spirals. Mathcad 
applications assume that the base of all spirals lies in the x-y 
plane and is centered at the origin, that the axis of all spirals 
is parallel to the z axis, and that the spirals are in free 
space. 
A. THE PLANAR SPIRAL ANTENNAS 

There are three major categories of planar spiral antennas: 
the eguiangular spiral, the Archimedean spiral, and the log- 
periodic spiral. The geometry of all planar spirals is pictured 
in Figure 7.1. 



67 




FIGURE 7.1 Planar Spiral Geometry 

In Figure 7.1, the spiral angle (B) is the angle between any 
radial line from the origin and a tangent to any edge of the 
spiral, (r) is the distance to any point on the spiral from the 
origin, and (r ) is the distance from the origin to the spiral's 
feed point. The Mathcad spiral antenna application user should 
not confuse the radial distance from the origin to any point on 
the spiral (r) and the distance from the origin to an observation 
point in the far-field (r ff ) . 

Spiral antennas may be constructed from wires or sheets of 
metal. For low power, receive only operations, spirals may also 
be built using printed circuit technology. A more rugged, all- 
purpose antenna is constructed by simply cutting the spiral edges 



68 



from a sheet of metal and running coaxial feed lines along the 
spiral arms. A dummy feed line may also be run on an opposing 
arm for symmetry [Ref 6: pp. 14-4 - 14-7]. 

The physical dimensions of the spiral arms determine the 
type of spiral antenna and the antenna's parameters. An 
eguiangular spiral is one whose edges or wires satisfy the 
following [Ref 5: p. 283]: 



i=z e a * (m) (7.1) 

In (7.1), (a) is an arbitrary constant called the flare rate. If 
the flare rate is a negative number, the spiral is considered 
left-handed. If the flare rate is positive, the spiral is right- 
handed. 

When sheet metal is used to construct an equiangular spiral, 
(7.1) defines the coordinates of one edge of one spiral arm. The 
next edge (r 2 ) is cut using the same spiral curve as (7.1), but 
with an angular arm width (6) as follows: 

r 2 = r e a{ *- b) (m) (7.2) 

Spiral antennas are usually symmetrical. Thus, for a two arm 
spiral, edges (r 3 ) and (rj are given by: 

r 3 =r e a, *- n) (m) (7.3) 

(7 A) 
r 4 = r e a( *-*- 6) (m) K ' ' 

Normally, flare rate is converted to a factor called 
expansion ratio (e ex ) which is written as [Ref 5: p. 284]: 



69 



e = r(<t> + 2rc) (dimensionless) (7.5) 

r(<|>) 

A typical value for the expansion ratio is 4 . 

The Archimedean spiral has many of the same characteristics 
as the equiangular spiral, except that any point on the edge of 
an Archimedean spiral is written as: 

r=r <$> tin) (7.6) 

Likewise, the performance of a log-periodic spiral is 
similar to an equiangular spiral except that its edges are 
defined by: 

r=r a* tin) (7.7) 

For a log-periodic spiral it can be shown that the following 

formula is always satisfied [Ref 2: pp. 697-698]: 

4> = tan(P)ln(r) (radians) (7.8) 

In (7.8), the spiral angle (B) is the same as that illustrated in 
Figure 7 . 1 and is constant at any point on the log-periodic 
structure. Because of their broadband characteristics and ease 
of construction, log-periodic spirals are regularly used in the 
lower millimeter wave region [Ref 12: p. 17-28]. 

The lowest operating frequency (f low ) , highest operating 
frequency (f hlgh ), and bandwidth (BW) of spirals are functions of 
the antenna's dimensions and the feed arrangement. For 
equiangular and Archimedean devices, the minimum radius (r ) and 
wavelength of the highest operating frequency are approximately 



70 



correlated as follows [Ref 5: pp. 284-285]: 

r o =J ^p {rn) (7 * 9) 

The wavelength (X low ) corresponding to the lowest operating 
frequency is set by the overall radius (R) of the structure as 
follows: 

R=h^ (m) (7.10) 

4 

With the antenna's upper and lower frequency limits 
established, bandwidth may be expressed as: 

BW=f high -f low (HZ) (7.11) 

The frequency limits and bandwidth of a log-periodic spiral 
are also determined by its dimensions. Bandwidth of a log 
periodic spiral is given by (7.11), where the lower frequency 
limit corresponds to the wavelength computed using (7.10) and the 
upper frequency's wavelength is calculated as follows [Ref 2: p. 
700] : 

hl±£» ( m) (7.12) 

° 20 

A spiral antenna is frequency independent in that most 

antenna parameters do not vary over the bandwidth of the antenna, 

which can be a considerable range of frequencies. Input 

impedance is one of these parameters and may be computed using 

the concept of complimentary antennas. The complement of a 

spiral is formed by replacing metal with air and air with metal. 



71 



The impedances of the spiral and its complement are (Z metal ) and 
(Z air ) , respectively. These two impedances are real, frequency 
independent, and for a two-arm spiral related as follows [Ref 5 
p. 283]: 



Z 



,^•4 <q 2 > <7 - i3) 



metaj. hj.z 

In (7.13), (?7 ) is the intrinsic impedance of free space. If the 
antenna and its complement are identical, the antenna is called 
self-complimentary and the angular arm width in (7.2) is tt/2. In 
the specific case of a two-arm, self-complimentary spiral, the 
impedance of the antenna is: 

S«e.j-2.ir=-Y=188.5 (Q) (7.14) 

Self-complimentary spirals are fairly common because they 
yield desirable radiation patterns. Measured input impedance is 
typically between 120-160 ohms for these devices, which is lower 
than the theoretical results of (7.14). The presence of a feed 
structure, the finite size of the antenna, and the finite 
thickness associated with the spiral's arms are the reasons 
measured input impedance is lower than its theoretical value [Ref 
5: pp. 285-286] . 

Mathcad spiral antenna applications assume that planar 
spirals are equiangular and self-complimentary. The applications 
also assume that input impedance for a two-arm spiral is given by 
(7.14) . 



72 



An (N) arm, rotationally symmetric spiral has (N-l) 
independent radiation modes, where each mode corresponds to a 
different radiation pattern. Typically, an (N) arm spiral 
operating in mode (M) excites each arm with a current of 
identical magnitude. The phase difference (a) with respect to 
the first arm for arm (n) of an (N) arm spiral excited in mode 
(M) is given by [Ref 6: p. 14-4]: 

-2nnM {radians) (7.15) 

N 

Most of the radiation from a spiral occurs at the point 
where the spiral's circumference (C) equals the product of the 
mode and wavelength of the frequency of interest. As frequency 
and wavelength change, the principal radiation point on the 
antenna changes but the radiation parameters and beam patterns do 
not. The shifting radiation point along the structure is why 
spirals are broadband antennas. In addition, as long as the 
spiral is sufficiently large to radiate all desired frequencies, 
the shape of the spiral arm termination does not effect the 
antenna's radiation pattern [Ref 6: pp. 14-4 - 14-7]. 

All modes of a planar spiral whose center is at the origin 
in the x-y plane have a null along the z axis, unless mode one is 
being excited. A planar spiral radiates in both positive and 
negative z half spaces with magnitude patterns that are 
rotationally symmetric. 

The electric field of a self-complimentary, planar spiral at 
an observation point in the far-field is given by [Ref 14: p. 

73 



530] : 



E Q =E r = (V/m) (7.16) 



E k 2 cos(6) [l+j'acos(e)]- 1 - J ' ( ^ /a) [tan (-?)]" 



e 



j(M(6--ZL) -kr ff ) 



sin(6) 2 (7.17) 



* e 

iff 



-jw* ( y//n) 



In (7.16) and (7.17), (E ) is a source strength constant and (k) 
is the wavenumber given by: 

k=~£- (m- 1 ) (7.18) 

A 

Rearranging (7.17), the amplitude of the electric field (A^) 
becomes : 

ft — tarT : (acos(6) ) 

cos(6) [tan(-^)]"e a (n ... 

2 / / « ( 7 . 19 ) 

V — W/77) 

sin (6) v /l+a 2 [cos (6) ] 2 

The Mathcad application user should note the phase variation 
as a function of azimuth in (7.17). This phase variation can 
result in azimuthal beam shaping if multiple spirals are used in 
an array [Ref 15: p. 18]. However, for a single spiral there is 
no change in electric field magnitude with azimuth and the 
pattern is considered rotationally symmetric. 

The electric field computed using (7.17) or (7.19) applies 
only in the far-field. Therefore, the following conditions must 
all hold if the far-field patterns are to be accurate [Ref 3: p. 
92] : 



74 



r ff >l.eX (m) (7.20) 

r ff zlOR (m) (7.21) 



8Rr 
X 



r ff z-2£- (m) (7.22) 



With the spiral's electric field given by (7.16) and 
(7.17), the antenna's radiation intensity (U) , radiated power 
(P raci ) , and directivity (D ) are found using generic antenna 

formulas as follows [Ref 3: pp. 28-29]: 



U= -IL\E J2 s _L_u J 2 (W/solid ana) (7.23) 

2ti 2ti 



P rad =ff Us in id) dddty (W) (7.24) 

AnU 

D max (dimensionless) (7.25) 

P 

iaa 

In (7.24), (n) is the entire sphere of radius (r ff ) in the far- 
field that surrounds the spiral. In (7.25), (U mav ) is the maximum 
radiation intensity anywhere on the sphere. 

The gain (G) of a spiral is the product of its directivity 
and efficiencies. Conduction-dielectric efficiency (e cd ) of a 
spiral cannot be easily determined and is, therefore, assumed by 
the Mathcad applications to be unity. Reflection efficiency 
(e rv ) , however, can be approximated. The impedance of a two arm, 
self -complimentary spiral is given by (7.14). In the general 
case of an (N) arm, self-complimentary spiral in free space 

75 



operating in mode (M) , the input impedance is real and is given 
by [Ref 6: p. 14-22] : 

Z .=N ^ (Q) ,„ „ x 

2 . . M, (7.26 

sin (ti — 

If the characteristic impedance (Z ) of the feed line is known or 
can be estimated, the voltage reflection coefficient (T) , 
reflection efficiency, and gain are computed by: 

Z ~Zr 

T= — - — - (dimensionless) (7.27) 

e rv =l-|r| 2 (dimensionless) (7.28) 

G=e zv D Q (dimensionless) (7.29) 

The planar spiral radiates in both directions normal to its 
surface. Improvements to antenna gain can be realized if the 
radiation in the undesired direction is reflected or eliminated. 
A common technique used to achieve an improvement in gain is to 
place a cylindrical metal cavity on the side of the spiral that 
has the unwanted beam pattern. The cavity can improve gain by up 
to 4.5 dB but can also reduce bandwidth by up to a factor of 5. 
The loss of bandwidth can be mitigated by filling the cavity with 
electromagnetic energy absorbing material. The absorbing 
material will reduce gain for spiral with a cavity by up to 1.5 
dB depending on thickness of the material and dimensions of the 
cavity. Although Mathcad applications do not include cavity 
parameters, the effect of cavities on antenna gain can be 

76 



estimated using measured results from an archimedean cavity- 
backed spiral. The change of overall gain with cavity depth and 
maximum gain with cavity diameter are pictured in Figures 7.2 and 
7.3, respectively [Ref 6: pp. 14-17 - 14-18]. 



o 



— e 



i o 



Effect of Cavity Diameter on Goin 



l7 ! I I I I 

- U- 1 ! I S ; 

- / : I : I i 

/: : III! 

iriizririi it:: 



. 3 O . -* . 5 O . e D . "7 Q . 5 

D i o m ete r / wavelength 



O S 



FIGURE 7.3 Effect of Cavity Diameter on Gain 



77 



o 

— 2 

— -i 

CO 

TO 

"15 

CO 

— S 

— 1 o 

c 


Effect of Cavity Depth on Gain 






















































O.I 0.2 0.3 O 

Cavit_> Dept^i in Wovelengtns 


4- 



FIGURE 7.2 Effect of Cavity Depth on Gain 



Effective isotropic radiated power (EIRP) is the product of 
gain (dimensionless) and power delivered to the input of the 
antenna. Effective isotropic radiated power for a planar spiral 
is given by: 



EIRP=P rad D (W) 



(7.30) 



If the magnitude of the current (I ) at the antenna's feed 
is known, its radiation resistance (R r ) may be expressed as: 



Kr = " 



2P 



iad 



I, 



(Q) 



(7.31) 



The spiral antenna Mathcad application user should note that the 
magnitude of the transmitter current cannot be estimated from the 



78 



antenna's dimensions and is normalized to unity. 

The polarization of a planar spiral whose center is the 
origin is circular within 70° of the z axis. The handedness of 
the spiral's polarization is the same as the spiral if measured 
in the +z half-space and if the spiral is excited at the central 
feed point. If the polarization is measured in the -z half-space 
or if the spiral is fed at its peripheral termination, the 
handedness of polarization is opposite that of the spiral. Dual 
polarization is achieved when the structure is simultaneously 
excited at both the center and periphery [Ref 6: p. 14-20]. 

Mathcad applications assume that a planar spiral's unit 
polarization vector (a a ) is circular, although the user may 
modify this according to observed feed structures. If the 
incoming wave unit polarization vector (a K ) can be determined, 
the polarization loss factor (PLF) of the antenna is calculated 
as [Ref 3: p. 51] : 

PLF=\a w -o' a \ 2 (dimensionless) (7.32) 

With all efficiency and loss terms computed, the antenna's 
maximum effective aperture (A em ) and maximum effective height 
(h em ) are estimated as follows [Ref 3: p. 63]: 

A em =e rv D (PLF) (Ji) (/T? 2 ) (7.33) 



h^pd- {m) (7.34) 



err, 



Figure 7.4 illustrates the difference between the electric 

79 



field pattern computed by the spiral antenna Mathcad applications 
and measured results for an equiangular spiral antenna (a=.3, 
<5=90°, r c =.005 meters, R=.142 meters) operating at 2 . 8 GHz over a 
conducting plane [Ref 16: p. 185]. 



Comparison of Equiangular Spiral Antenna Electric Fields 





o.s 




0.6 


E 




c 


0.4 


c 




-a 


0.2 


TZL 




&> 




a 





^ 




■a 




c 


-0.2 


L_ 




O 




— 


-0.4 


u 




v— 




LlI 






-0.6 



-0.8 - 



-1 



..-■■"' // ' C! a I .? uj.d.t§.d _ ElectKjc Field 
''■/^•■•'Measured DiSctrj.cX -Field '"■••.. 


..••'"'•■.... / ""; 


x \ ...•"'"'••... 



FIGURE 7.4 Equiangular Spiral Electric Field Patterns 

B. THE CONICAL SPIRAL ANTENNA 

The planar spiral antenna offers many features which make it 
a very popular device. However, the fact that it radiates in a 
direction normal to both of its surfaces is a major drawback. 



80 



Although cylindrical cavities placed on one side of the spiral 
can reduce the effects of unwanted radiation, a modification to 
the basic planar structure can accomplish the same result without 
additional apparatus. If the planar spiral is altered into a 
conical shape, many of the desirable features of the planar 
spiral are maintained, but radiation occurs primarily in 
direction of the cone's tip. 

Any point on the i th edge of a log-conical spiral antenna 
may be defined by [Ref 5: p. 286]: 



^i=^o e 



U*inB H*-»i> (/n) (7.35) 



In (7.35), (6.) is the angular offset of the i th edge, (0 C ) is the 
conical half-angle, and (b) is an arbitrary constant given by: 

£>=cot(P) {dimensionless) (7.36) 

The geometry of a log-spiral is illustrated in Figure 7.5. 



81 




FIGURE 7.5 Conical Log-Spiral Geometry 

In Figure 7.5, (B) is the overall diameter or twice the overall 
radius (R) and (d) is twice the feed radius (r ) . 

Analogous to the planar spiral, the upper operating 
frequency (f high ) of a conical log-spiral is determined by the 
relationship between the wavelength (A, high ) of the upper frequency 
and the spacing (d) between feed points as follows [Ref 5: p. 



82 



287] : 



d=h±sH ( m ) (7.37) 



The lower operating frequency (f iOK ) of the conical spiral is 
determined by the correlation between the lower frequency's 
wavelength (A iow ) and the antenna's overall diameter by [Ref 5: p. 

287] : 

B= 2how (m) (7.38) 

8 

The bandwidth of the conical log-spiral is computed using 
(7.11) . 

The conical log-spiral radiates a single lobe in the 
direction of its apex. The pattern broadens with increasing 
spiral angle (B) and lowering cone angle (28 ) until 
irregularities occur and multiple beams begin to form. Figure 
7.6 provides a rapid reference for the boundary between usable 
and unusable cone and spiral angles. [Ref 13: pp. 9-84 - 9-85] 



83 



Useable Conical l_oa — Spiral Dimensions 





1 1 


Useable F?eo> 


o n ■ 




j 






j 






1 


















/ 


OnM»»at>lw Reglo^ 


f 

/ 







iciudeO Come Angle 



FIGURE 7.6 Conical Log-Spiral Useable Dimensions 



The far-field electric field components of a conical log- 
spiral of total arm length (L) are written as follows [Ref 17: 
pp. 321-331] : 



E z = (V/m) 



(7.39) 



~-Jkz [c rL -ZM C os(6)cos(e D ) 



A(Z)c% (V/m) (7.40) 



2r ff QJo 



i^cos(e)cos(6 ) 

P BiDdi (V/m) (7.41) 



In (7.40) and (7.41), the slowness factor of the antenna (Q) and 
the total arm length of a spiral arm (L) are constants defined 
by: 



84 



Q= K 1+ — {diwensionless) (7.42) 

N b 2 



L ^l£[ e ( ^ )bsiri{ ^ ] -1] (w) (7.43) 

b 

In (7.43), (0 L ) is the azimuth at the end of the spiral arm and 

is: 

R 



1 sin(0 o ) (7.44) 

<l>r=-r— = — rs"^ ln( — ) (radians) 

The coefficients A(£) and B(£) of equations (7.40) and (7.41), 
respectively, are expressed as: 

sin(e o )cos(0)C(O . (7 ' 45) 

•[ — -sin (6) cos (0 O ) ] 

(dimensionless) 



(7.46) 



B(l) =Jf~ 1 e - iWe i f s -^ sin < e o)cos( ( |»(0-*-ia) 

-J7sin(6 ) 
*( —)C(£) (diwensionless) 

In (7.45) and (7.46), (a) and C(£) are given by: 



2 71 

a = -^— (dimensionless) (7.47) 



C(0 = (l^-^-)e^^^ , i(l--4 FT ) ,, ._. 
jbsm(0 o ) jbsm(6 ) (7.48) 

* e -ji4U)-*+i*) diwensionless 
The ( + ) sign in (7.48) is for C($) used in calculating (E 6 ) , 
while the (-) sign is for (E^) computations. 

85 



Azimuth angle as a function of distance along the spiral arm 
(0(£)) in (7.45), (7.46), and (7.48) is calculated using: 

4>(S> = u • 1 , a v ln(-^-H) (radians) (7.49) 

jbsin(0 o ) r 

The current distribution (I(£)) used in (7.40) and (7.41) is 
that of the excitation current along the spiral arm. This 
current distribution can only be calculated using intricate 
numerical modeling or a method of moments solution. An 
examination of measured current distribution yields the following 
engineering approximation of (l(£)) given a current at the feed 
point of (I ) : 

I(Z,)=I e~~ I (Amps) (7.50) 

Figure 7.7 provides a comparison of the current distribution 
computed by the conical log-spiral Mathcad applications using 
(7.50) to that measured for a four arm conical log-spiral 
(0 O =1O°, 6=20 c , b=.46) operating in mode one. The application 
user should note that most of the radiation from the conical log- 
spiral occurs at the location on the device where arm length (L) 
in wavelengths approximately equals the mode number [Ref 17: pp. 
321-331] . 



86 



C o m p a ri s o n i of Conical Log — Spiral Current Di st r-I fc> i_i"t io 



"7 



J - 































- Calculated Curre 


nt 






\ — h 


;< ea s u red 


C u rre nt 
















































i — : -— ^-— 



o 



12 3 4-5 

Spiral Length in Wavelengths 



FIGURE 7.7 Conical Log-Spiral Current Distributions 

Mathcad spiral antenna application users should note the 
precise evaluation of the integrals in (7.40) and (7.41) is very 
time consuming, even on a 33 MHz, 386 personal computer. Thus, 
the application user is provided a trapezoidal approximation to 
evaluate the conical log-spiral antenna integrals. Also, both 
the exact and trapezoidal applications only analyze electric 
fields and other parameters in the +z half- plane. 

In the trapezoidal approximation, the integral is replaced 
by a summation and (d£) is replaced by: 



87 



d£«- (m) (V.51) 

In (7.51), (t) is an operator entered number of spiral arm 
increments. The application user should expect the Mathcad 
evaluation of a conical log-spiral using the trapezoidal method 
to take about 2 5% of the time required for exact method 
predictions. 

Conical log-spiral electric field calculations are valid 
only if the conditions of (7.20) through (7.22) are satisfied. 
Mathcad applications assume that overall radius is the largest 
dimension of the conical log-spiral. 

Radiation intensity of the conical log-spiral antenna is 
determined by [Ref 3: p. 28]: 

[/=£££_ [\e q \ 2 + \e^\ 2 ] (W/ solid ang) (7.52) 

The radiated power and directivity of the conical log-spiral 
are given by (7.24) and (7.25). Users of the Mathcad spiral 
antenna application should be aware that computer solution of 
these formulas may be very time consuming. However, one may 
reasonably consider most conical log-spiral antennas in the 
useable region of Figure 7.6 to have a total electric field that 
is rotationally symmetric with respect to the antenna's axis. 
Given this fact, an approximation of directivity may be made. An 
estimate of half-power beamwidth (A6) in degrees as a function of 
spiral and cone angles is provided in Figure 7.8 below. 



88 



I alt Power Beomwidth as a Function of Conical Log — Spiral Geo nn 4 



22; 



200 



1 80 



1 60 



1 4-Q - 



= 1 OO - 



SO 



SO 



4-0 




*o 



SO 



60 70 

Spiral Angle 



SO 



90 



FIGURE 7.8 Conical Log-Spiral Half_Power Beamwidths 

The half -power beamwidth from Figure 7.8 is used to estimate 
directivity by [Ref 13: p. 9-87]: 



D„* 32600 (dB) 



(A6) 2 



(7.53) 



It is impossible to accurately determine the conduction- 
dielectric efficiency of a conical log-spiral antenna based 
solely on measured geometry of the antenna. Hence, the Mathcad 
conical log-spiral applications assume all conduction dielectric 
efficiency is unity. The input impedance (ZJ of a conical log- 
spiral antenna as a function of angular arm width is provided in 
Figure 7.9. 



89 



J50 

JSOO 

250 
E 

6 200 
"E 

g- 

— T SO 
1 OO 


Input I r-n ped n c e 0^ a Conical Log — Spiral 




i ! 






\ \ \ 30' Deg 
N, \.^*?' D e 9 




nST I 




Cone Angle — 15 TP-eg^^ 








7/ 


^\\ 


O 50 1 OO 1 SO 200 

A .n c. _j 1 r ArrT, Width in Degrees 



FIGURE 7.9 Input Impedance of a Conical Log-Spiral 



Using the input impedance from Figure 7.9 and an estimate of the 
characteristic impedance of the feed assembly, the Mathcad 
conical log-spiral antenna application user may compute voltage 
reflection coefficient, reflection efficiency, gain, and 
effective isotropic radiated power using (7.27), (7.28), (7.29), 
and (7.30), respectively. 

If the polarization unit vector (a w ) of an incoming wave is 
known, precise determination of polarization loss at a point in 
the far-field may be desired. To accomplish this, Mathcad 
applications convert electric field components of (7.40) and 
(7.41) to Cartesian coordinates at a user defined point (x,y,z) 
and compute the antenna's unit polarization vector (a a ) as 



90 



follows [Ref 4: pp. 35-36, 364-367]: 



£ =£ e cos (6) sin(4>) -E 4 sin (<t>) (V/m) 



E y =E e cos (8) sin (4)) +^cos (<|>) ( V/m) 



£ 2 = -£; e sin(8) (V/m) 



aj£ +3JZ +a E 
o a (x,y, z) = — ^— - — — - (dimensionless) 

y/\E(x,y,z) | 2 



(7.54) 
(7.55) 
(7.56) 

(7.57) 



The conical log-spiral antenna's polarization loss factor is 
calculated using (7.32). Using the polarization loss factor, 
maximum effective aperture (A e: J and maximum effective height 
(h em ) for a conical log-spiral are given by: 



A m ^ X vD (-^)PLF (m 2 ) 



(7.58) 



h err =2 



*A» 



\ ^ 



\m) 



(7.59) 



Figure 7.10 and Table 7.1 compare measured data to that 
calculated by the Mathcad applications for a two arm, mode one 
conical log-spiral (b=.053, 6 o =10°, a=73°, d=.03 meters, B=.30 
meters, 6=90°) operating at 350 MHz [Ref 18: p. 332]. 



91 



Comparison of Conical Log — Spiral Antenna Electric Fields 





0.6 




0.6 


E 


0.4 




0.2 


a; 
i_ 





o 




i_ 


-0.2 






Ld 


-0.4 




-0.6 




-0.5 




-1 




Calci>lJpted\Fie|.d.J 

'■Measured Fi-e'fcK 



••-..\ 



ijpi!. 




FIGURE 7.10 Conical Log-Spiral Electric Field Pattern 



TABLE 7.1 Conical Log-Spiral Data Comparison 



ANTENNA 
PARAMETER 


MEASURED 
DATA 


CALCULATED 
DATA 


DIRECTIVITY 


6.10 dB 


6.03 dB 



92 



VIII. THE CONICAL HORN ANTENNA 

The conical horn antenna is a device that provides a 
transition for an electromagnetic wave from a circular waveguide 
to an unbounded medium such that the wavefront phase at the 
horn's aperture is nearly constant [Ref 2: pp. 644-645]. As a 
result of its axial symmetry, the conical horn's radiation 
pattern is strictly a function of the cone's geometry for a given 
mode of excitation [Ref 10: p. 10-3]. The conical horn is widely 
employed as a feed element for reflector assemblies used in 
satellite tracking, microwave communications, and radar. The 
geometry of a conical horn is illustrated in Figure 8.1. 



< 


y 












s^ ^\ ' 


i 








-. <M 


( A\ 


d 


f 






L 








1 


- Z 




N 


° / 

' / 










h/ 


/ 




u 




\ / 


x X I 




1 


■ 


CIRCULAR 
WAVEGUIDE 




t 

MOUTH OF 
HORN 



FIGURE 8.1 Conical Horn Geometry 

In Figure 8.1, (a) is the inner radius of the circular waveguide, 
flare angle (a) is the included angle of the horn, (d) is the 



93 



diameter of the mouth of the horn, and the axial height (h) is 
the distance from the origin to the center of the mouth of the 
horn. 

A circular waveguide will only propagate a transverse 
electric (TE) or transverse magnetic (TM) mode of an 
electromagnetic wave if the freguency of the wave is above a 
minimum value for the mode called the cutoff freguency (f c ) . The 
propagating mode with the lowest cutoff freguency is called the 
dominant mode. For circular waveguides the (TE n ) mode is the 
dominant mode [Ref 4: p. 570]. Mathcad conical horn antenna 
applications assume that the waveguide is excited only in the 
dominant (TE n ) mode. Also, the applications do not compute 
bandwidth for conical horn antennas. Rather, the software 
computes cutoff freguencies for selected modes such that the user 
can determine if the waveguide will support propagation of a 
specific mode. 

Each mode's cutoff freguency is a function of the circular 
waveguide's inner radius. For transverse electric waves, the 
cutoff freguency of an air filled circular waveguide is given by 
[Ref 19: pp. 472-473] : 

^cK.n=—^-= &z) (8.1) 

In (8.1), (n ) is the permeability of free space, (e ) is 
permittivity of free space, and (x'mr.) i s the nth zero of the 
derivative with respect to the argument of the Bessel function of 



94 



the first kind, order m. (x'mn) ma y be obtained from Table 8.1 



TABLE 8.1 Zeroes of the Bessel Function Derivative 



X mn 


m=0 


m=l 


m=2 


m=3 


m=4 


n=l 


3.8318 


1.8412 


3.0542 


4.2012 


5.3175 


n=2 


7.0156 


5.3315 


6.7062 


8.0153 


9.2824 


n=3 


10.1735 


8.5363 


9.9695 


11.3459 


12.6819 


n=4 


13.3237 


11.7060 


13.1704 


14.5859 


15.9641 



The cutoff frequency of transverse magnetic waves in an air 
filled circular waveguide is written as [Ref 19: pp. 478-479]: 



^ c' n, n 



(Hz] 



2na y /\i e l 



(8.2) 



In (8.2), (XmrJ i- s tne nth zero of the Bessel function of the 
first kind, order m. (XmiJ may be obtained from Table 8.2. 

TABLE 8.2 Zeros of the Bessel Function 



Amn 


m=0 


m=l 


m=2 


m=3 


m=4 


n=l 


2.4049 


3.8318 


5.1357 


6.3802 


7.5884 


n=2 


5.5201 


7. 1056 


8.4173 


9.7610 


11.0647 


n=3 


8.6537 


10. 1735 


11.6199 


13.0152 


14.3726 


n=4 


11.7915 


13. 3237 


14 .7960 


16.2235 


17.6160 



95 



A conical horn is said to be optimum if the diameter of the 
mouth of the horn satisfies the following [Ref 10: p. 10-9]: 

d=-Xsin( — ) (m) (8.3) 

To allow for comparison between the antenna being evaluated and 
an optimum horn, the conical horn Mathcad applications calculate 
optimum (d) for a user provided frequency of interest and horn 
flare angle. 

The electric field calculations of the conical horn Mathcad 
applications use the magnetic field integral equation solution 
for aperture antennas. This is similar to the combined field 
integral equation solution first postulated by Schorr and Beck 
for conical horns in 1950 [Ref 20: p. 795]. The application user 
can expect a large amount of computer processing time will be 
necessary to analyze a conical horn and that this time will grow 
with the square of the number of increments (i) into which the 
far-field is divided. 

In order to compute the magnetic vector potential (A) of the 
aperture field of a conical horn, several preliminary functions 
and related constants must first be defined. These include 
associated Legendre functions of the first kind of order (v) 
[P„ (m) cos (6) ] , spherical Hankel functions of the second kind of 
order (v) [hj 2) (kr) ] , and the derivative with respect to (kr) of 
the spherical Hankel functions [h'„ C2) fkr) ] . The arguments of the 
spherical Hankel functions are (r) , the distance from the origin 
to an observation point in the far-field, and the wavenumber (k) 

96 



for a wavelength (X) corresponding to a frequency of interest. 
Wavenumber is given by: 

k=M (w- 1 ) (8.4) 

The order (v) of the Legendre and Hankel functions may be 
approximated using the constant (b ) as follows [Ref 21: p. 521]: 

(1--*) 
log[ -2— ] , R . 

b = {dimensionless) 

log [cos ( — ) ] 

v = - . 5 + . 5./l+4i:v (dimensionless) (8.6) 

The associated Legendre function of the first kind with m=l 
may be estimated using the gamma function (T(z)) for angles less 
than (7T/3) as follows [Ref 22: pp. 336]: 



P v (li (cos(6))= r(v * 2 > U ^sin(6)3cos((v + . 5)6 + 4) r« 7^ 

r(v+l . 5) \ 2 4 (8.7) 

( dimensi onl ess) 
The value of the spherical Hankel function and its 
derivative with respect to its argument in the far-field may be 
estimated by [Ref 17 p. 796]: 

<7\ 1 -j(ki- (v ! 1) n) , c QN 

iV (£*) - — e 2 (dimensionless) («>.b) 



i: v (2) - (jcr) =— e" J(Jcr " 2 * [i-j (v+1) ] (dimensionless) ( 8 -9) 
Jcr Tcr 



The components of the magnetic vector potential are given by 



97 



[Ref 20: p. 798] : 



a __n f 2 f * ~jkhcos (P) 
* "Jo Jo 
* [sin (60' )P V (cos(0 o ) )cos 2 (<t>' )cos(6' ) 

+ 6cos(69') (cos (e } } gin2 (<J)1 j sin2 (e , } ] 
sin (0 ' ) ° 

* c?6 ' c?4> ' ( Mb/m) 

* [sin (66' )P v (cos(6 ) ) cos (<t> ' ) sin (<J> ' )cos(0' ) 

5cos(60') (CQS (0 } } sin ((j) , ) cos (e , ) sin 2 (e , } ] 
sm (0 ' ) ° 

*d0'Q'4)' (Wb/m) ) 



(8.10) 



(8.11) 



Jl __g f 2 C zn e Jkhcos <p: 

z "Jo Jo (8.12) 

* [sin(60')P v (cos(0 o ) ) cos (4)') sin (0' ) ] dQ ' c?4> ■ 

( Wb/m) 
The integrals in (8.10) through (8.12) are performed over the 
aperture of the horn, thus the primed components in these 
integrals indicate source coordinates. The terms cos(B), (£), 
and (B H ) in the magnetic vector potential integrals are given by 

cos ((3) =cos (0)cos (8' ) +sin(8) sin(8' ) cos ((t>-4>' ) 

(8.13) 

( dimensi onl ess) 



b- — (dimensionless) (8.14) 

a 



B,.=B - e - jkr [\h- 2) (kh) +kh, 2) ' (kh)] (Wb/m) (8.15) 

j(2Ti) z fe r h 



In (8.15), (BJ is an amplitude scaling constant. The magnetic 
vector potential integrals contained in (8.10) through 

98 



(8.12) include approximations of the associated Legendre function 
and its first derivative with respect to cos(0') as follows [Ref 
20: p. 798]: 

P v (1) cos (6' ) ~sin(60' ) P v (1) cos ( — ) (dimensionless) (8.16) 

P t (1), C08(6')- dC ? 8 |y.' ) P 1> (1) COB(-|) (Q 17 , 

sm(0') 2 (8.1/) 

(dimensionless) 

Far-field electric field components are approximated by 
converting the magnetic vector potential components to spherical 
coordinates and multiplying by (k 2 ) as follows [Ref 3: pp. 92, 
102] : 

A e =A x cos (0) cos (<t>) +A y cos (0) sin(<|>) -A z sin(Q) (Wb/m) (8.18) 

^ = -A x sin(4>) +A y cos (<J>) (Wb/m) (8.19) 

E Q =k 2 A^ (V/m) (8.20) 

E^=k 2 A^ (V/m) (8.21) 

For the electric field components of (8.20) and (8.21) to be 
valid, the observation point (r) must be in the far-field. 
Therefore, assuming the diameter of the mouth of the horn is the 
largest dimension of the cone, all of the following formulas must 
be satisfied [Ref 3: p. 92]: 



99 



rzl.6\ (m) (8.22) 

rz5d (/7?) (8.23) 



r *2^! (m) (8.24) 



When investigating the parameters of a conical horn antenna 
with the magnetic field integral equation, it must be presumed 
that no electric field exists behind the mouth of the horn. 
Thus, radiation intensity (U) , radiated power (P rad ) , and 
directivity (D ) of the conical horn antenna are calculated by 
applying generic antenna formulas as follows: 



U=-^- [\E Q \ 2 + \E^\ 2 ] (W/ solid ang) (8.25) 



271 



o 



71 

P Tad=f ZT [ ~* Us in (Q)ded<b (W) (8.26) 

J Q JO 



4 it £7 

D max (dimensionless) (8.27) 

p 



In (8.25), (T7 ) is the intrinsic impedance of free space and in 
(8.26), (U max ) is the maximum radiation intensity from the 
antenna. For a conical horn, it may be reasonably assumed that 
maximum radiation intensity occurs along the antenna's axis 
(i.e. , 6=0, <p=0) . 

In order to lower the computer processing time of the 
conical horn Mathcad applications, the user may choose a file 
that uses a trapezoidal approximation of the integrals in (8.10) 



100 



through (8.12) and (8.26). The user may also vary the number of 
trapezoidal increments (tl, t2 , t3) to vary the extent of this 

approximation and to adjust computation time. 

The ohmic losses of conical horns and their associated 
circular waveguides are very difficult to precisely determine, 
but are normally very small. Likewise, reflection efficiency of 
a conical horn cannot be easily found using analytical 
techniques, but when measured for horns with moderate flare 
angles and high directivity it is usually close to unity. The 
conical horn antenna Mathcad applications approximate the product 
of ohmic and reflection efficiencies as 0.95 and compute antenna 
gain (G) and effective isotropic radiated power (EIRP) by [Ref 
10: p. 10-9] : 

G=.9 5D (dimensionless) (8.28) 

EIRP=P xad D (W) (8.29) 

The unit polarization vector (a a ) in Cartesian coordinates 
of a conical horn's wave may be computed at any point in the far- 
field as follows [Ref 4: pp. 35-40, 364-367]: 

^=£ e cos (0) cos (4>) -£' 4) sin((})) ( V/m) (8.30) 

E v =E e cos(Q) sin (4>) +E i> cos(<$>) (V/m) (8.31) 

E. = -E a sin{Q) (V/m) (8.32) 



a E + a £ +a E 
o a (x,y, z) = x x — £-£ — z -^- (dimensionless) (8.33) 

y/\E(x,y,z) | 2 



101 



If an incoming wave's unit polarization vector (a w ) is known 
or can be estimated, the polarization loss factor (PLF) of the 
antenna at a point in the far-field can be expressed as [Ref 3: 
p. 51]: 

PLF= |o w ,-o*| 2 (dimensionless) (8.34) 

The maximum effective aperture of a conical horn may be 
determined as follows [Ref 3: p. 63]: 

A=.95D (PLF) (-£1) (m 2 ) (8.35) 

em o 4 ^ 

The actual value of current (I D ) applied to the input of a 
conical horn cannot be determined by dimensional information 
alone. Thus, exact calculation of radiation resistance (R r ) is 
impossible. However, if input current is normalized to one amp, 
normalized radiation resistance and maximum effective height 
(h erL ) may be written as [Ref 2: p. 42]: 

^r = T^TT (Q) (8.36) 

\lo\ 2 



Kr = 2. 



RrKjn {m) (8.37) 



A conical horn is a member of a group of devices known as 
aperture antennas. A term frequently used to analyze the 
performance of aperture antennas is aperture efficiency (e ap ). 
Aperture efficiency is the ratio of maximum effective aperture to 
physical area at the mouth of the horn and is calculated by [Ref 



102 



3: p. 475]: 



e *P= 



,(|)» 



{dimensionless) 



(8.38) 



Aperture efficiency of a conical horn is typically about 50%. 

Table 8.3 and Figure 8.2 compare measured data to that 
calculated by the Mathcad applications for a conical horn (a=.045 
meters, h=1.489 meters, d=1.0 meters, and a=20°) operating at 
1.96 GHz [Ref 22: p. 100]. 

TABLE 8.3 Conical Horn Data Comparison 



ANTENNA 
PARAMETER 


MEASURED 
DATA 


CALCULATED 
DATA 


GAIN 


22.0 dB 


2 3.6 dB 



Com p a riso n of Conical Horn Antenna Electric Fields 



1 
D.S 

o.e 

0.2 

O 
•0.2 

■O.A- 

■o.e 
o.e. 
— i 




•^"'Caicuiote'd Electric Fieic 
— ' Measur.ea Eiee-tric Fie 



FIGURE 8.2 Conical Horn E-Plane Electric Field Patterns 



103 



IX. THE PYRAMIDAL HORN ANTENNA 

The most popular member of the horn family of antennas is 
the pyramidal horn. The pyramidal horn provides a transition 
from a rectangular waveguide to free space and is flared both 
horizontally and vertically. It possesses a radiation pattern 
that is essentially the combination of that of the E- and H-plane 
sectoral horns. If accurately constructed, the pyramidal horn 
offers the ability to control beamwidth in both principal planes 
with a gain (G) that closely matches theoretical predictions [Ref 
10: p. 10-3]. The geometry of the pyramidal horn in the E-plane 
(0=7r/2) and H-plane (0=0) is illustrated in Figures 9.1 and 9.2, 
respectively. 



+y 




RECTANGULAR 
WAVEGUIDE 



MOUTH 
OF HORN 



FIGURE 9.1 E-Plane Pyramidal Horn Geometry 



104 



+x 




t 

RECTANGULAR 
WAVEGUIDE 



Ph 



a, 



+Z 

MOUTH 
OF HORN 



FIGURE 9.2 H-Plane Pyramidal Horn Geometry 

In Figures 9.1 and 9.2, (a) and (b) are the dimensions of 
the rectangular waveguide used to excite the horn, (a x ) and (b x ) 
are the dimensions of the mouth of the horn, (ty erh ) are half the 
flare angles in the indicated planes, (Pi, 2 ,e,h) are tne distances 
from the imaginary apex of horn to the indicated points on the 
perimeter of the mouth of the horn, and (p eh ) are the distances 
from the beginning of the horn's flare to the center of the mouth 
of the horn along the indicated axis. 

To physically construct a pyramidal horn, the parameters 
(p e ) and (p h ) should be equal. These dimensions are calculated 
as follows [Ref 3: p. 568]: 



105 



p e ={b x -b) 



(£> 2 --i (») (9>1) 



p h =(a 1 -a] 



(£*)*-l (/n) (9.2) 

N a i 4 



The Mathcad pyramidal horn applications do not calculate a 
bandwidth of operating frequencies. Rather, the applications 
compute a matrix of transverse electric (TE) and transverse 
magnetic (TM) cutoff frequencies (f c ) based on the dimension of 
the waveguide. The cutoff frequencies are the lowest frequencies 
of a given mode which can propagate in the waveguide. Cutoff 
frequency for transverse electric or transverse magnetic wave of 
mode (m,n) in an air filled waveguide is given by [Ref 4: p. 
549] : 



if. 



2/^ N a h 



(iL')2 +( ^)2 {Hz) (9.3) 



In (9.3), (fi ) is the permeability of free space, (e ) is the 
permittivity of free space, and (m) and (n) are integers. For 
transverse magnetic waves, (m) and (n) must be non-zero. For 
transverse electric waves, either (m) or (n) may be zero, but not 
both. The mode with the lowest cutoff frequency is called the 
dominant mode. The (TE 10 ) mode is dominant in a rectangular 
waveguide. 

Assuming the electric fields behind the mouth of the horn 
are zero and the physical dimensions of the horn are negligible 



106 



in the far-field, one may express the electric field components 
of a pyramidal horn in the +z half-plane as [Ref 3: pp. 565-578]: 

E s = i kE °^r~ [sin( 4>) (l+cos(6) )I 1 I-J (V/m) (9.4) 

E,=jkE - A [cos ((J)) (l+cos(6)) I X I 2 ] (V/m) (9.5) 

In (9.4) and (9.5), (EJ is an electric field amplitude scale 
factor set to unity by the Mathcad applications, (r) is the 
distance from the origin to an observation point in the far-field 
and (k) is the wavenumber corresponding to the wavelength (X) of 
a frequency (f) of interest and is given by: 

jc=i_E. (,71-1) (9.6) 

The functions (IJ and (I 2 ) in (9.4) and (9.5) are computed by: 

/. 4 1 n -J>!-jp- -Ssin(6)cos(<J>)] 

-Z"i = l 1 cos( — ^e Pi dE, (dimensionless) (9-7) 

J - — - c2 



I. f2 

. -2 -jkl-±—-isin(6'sin{4>)] 

J 2 =J ^e " Pl dl (dimensionless) (9.8) 

2 

The Mathcad application user should note that closed form 
solutions of (9.4) and (9.5) exist but are very complex and 
tedious. Thus, the applications use a numerical approximation of 
these integrals to compute electric field components for the 
horn. 

The electric field components of (9.4) and (9.5) are 
applicable only in the far-field of the horn. Therefore, the 

107 



distance to the far-field observation point must satisfy all of 
the following [Ref 3: p. 92]: 

ZZ1.6X (m) (9.9) 

i^SD (m) (9.10) 

r^-^ (m) (9.11) 

In (9.10) and (9.11), (D) is the distance between opposite 
corners of the mouth of the horn. 

The radiation intensity (U) , radiated power (P rad ) , 
directivity (D ) , and effective isotropic radiated power (EIRP) 
of a pyramidal horn are calculated using generic antenna 
equations as follows: 

U=-^-[\E Q \ 2 *\E^\ 2 ] {W/ solid ang) (9.12) 

p rad=f{ Usin(6)dQd$ (w) (9-13) 



4 ti jj 
D o= — (dimensionless) (9-14) 

?! ad 



EIRP=P zad D (W) (9.15) 

In (9.12), (?7 ) is the intrinsic impedance of free space, in 
(9.13), (fi) is the hemisphere of radius (r) in the +z half-space, 
and in (9.14), (U m£X ) is the maximum radiation intensity anywhere 
on that hemisphere. 

The efficiency of a pyramidal horn is difficult to predict 

108 



based on measured geometry alone. However, the typical horn can 
reasonably be assumed to be approximately 50% efficient. Thus, 
Mathcad pyramidal horn applications compute gain as follows: 

G=.5D (dimensionless) (9.16) 

The unit polarization vector (a a ) of a pyramidal horn's 
electromagnetic wave at a point in the far-field is determined 
using the electric field components from (9.4) and (9.5). The 
antenna's unit polarization vector in Cartesian coordinates is 

computed by [Ref 4: pp. 34, 364-367]: 

£ x =£" 8 cos (6) cos (4>) -£^sin(<J>) (V/m) (9.17) 

E y =E e cos (0) sin ((J)) +^cos (4>) (v/m) (9.18) 

E z =-E 6 sin(Q) (V/m) (9.19) 

a E + a E +a E 
o a (x,y , z) = x x — ) —± — z —^ (dimensionless) (9.20) 

y/\E(x,y,z) | 

If an incoming wave's unit polarization vector (a w ) is known 
or can be estimated, the pyramidal horn's polarization loss 
factor (PLF) can be determined by [Ref 3: p. 51]: 

PLF=\o w -o' a \ 2 (dimensionless) (9.21) 

The maximum effective aperture (A em ) and aperture efficiency 
(e ap ) of the pyramidal horn is written as [Ref 3: p. 63]: 

A en ,= .5D Q (PLF) (2L) (m 2 ) (9.22) 



109 



e a = — v 2 - (diwensionless) (9.23) 



The amplitude of the input current (I ) used to excite the 
rectangular waveguide cannot be calculated based on dimensional 
information alone. Thus, in order to compute radiation 
resistance (R r ) and maximum effective height (h em ) , a normalized 
value of 1 amp is assumed to excite the waveguide. These 
parameters are expressed as [Ref 2: p. 42]: 

R= — -^ (0) (9.24) 



K^- 



^ {m) (9-25) 



The Mathcad pyramidal horn applications may also be used to 
analyze E-plane and H-plane sectoral horns. The term (£ 2 / 2 P) in 
the exponents of the integrals of (9.7) and (9.8) is a phase 
error term that accounts for differences in phase between the 
center and any point in the aperture of the horn. The sectoral 
horns are evaluated by eliminating the phase error term in the 
direction that is not flared. To accomplish this modification to 
(9.7) and (9.8), the application user is directed to set (al) 
egual to (a) for E-plane sectoral horn analysis and (bl) equal to 
(b) for H-plane sectoral horn analysis. If the application user 
makes these selections, (I 2 ) for the E-plane sectoral horn or 
(I 2 ) for the H-plane sectoral horn is altered, respectively, as 



110 



follows [Ref 3: pp. 536, 552]: 



A- 



-(-¥) [ 



cos ( — sin(6)cos(4)) ) 
2 



((^sin(6)cos((J))) 2 -(^) 2 ) 
(di/7?ei2sio.nies.s) 



(9.26) 



!,*[ 



sin( — sin(6)sin(4>) ) 
— sin(6)sin(4>) 



] (dimensionless) 



(9.27) 



Table 9.1, Figure 9.3, and Figure 9.4 compare measured data 
to that calculated by the Mathcad applications for a pyramidal 
horn (p!=.3398 meters p 2 =.3198 meters, a : =.1846 meters, b 1 =.1455 
meters, a=. 02286 meters, b=. 01016 meters) operating at 9.3 GHz 
[Ref 5: pp. 413-415] . 

TABLE 9.1 Pyramidal Horn Data Comparison 



ANTENNA 
PARAMETER 


MEASURED 
DATA 


CALCULATED 
DATA 


DIRECTIVITY 


21.3 dB 


21.6 dB 



111 



Comparison of Pyramidal Horn Electric Fields 

\ /i\\ — CatcuToted Electric Fielc 

..-•■'"'•. \ / ■ 'V Me'asu red. '"El.ectric Field 



^ : :*fcrprf 



FIGURE 9.3 E-Plane Pyramidal Horn Electric Field Pattern 



Comparison of Pyramidal Horn Electric Field Patterns 

1 



O.B - 

o.e - 

0.-4 

0.2 
O 

— 0.2 

— 0.4- 

— 0.6 

— O.B 

— 1 



...J i.\ — -■' Calculated E ectric Fifld 

"M-e.a-s i_i red Electric Field 




::: *V, f-; :'.'.... .i. 



FIGURE 9.4 H-Plane Pyramidal Horn Electric Field Pattern 



112 



X. REMARKS AND CONCLUSIONS 

The intent of this thesis and associated software was to 
provide the Naval Maritime Intelligence Center (NAVMARINTCEN) 
with a relatively simple, user friendly set of Mathcad 
applications that would analyze various types of antennas based 
solely on dimensional information and ground characteristics. 
Although the software format achieves the goal of easy use, the 
nature of the formulas applicable to many of the antennas 
necessarily reduces the simplicity of the programs. Indeed, many 
of the equations used in this research project are so complex 
that they are not found in any existing textbook dealing with 
antennas. Nevertheless, the Mathcad applications developed in 
conjunction with this thesis allow the user to analyze several 
antennas that cannot be studied with current electromagnetics 
software packages such as ELNEC, NEC, or WIRE. And, in every 
case, the Mathcad applications compute antenna parameters and 
far-field radiation patterns that closely compare with measured 
or predicted results. 

The Mathcad applications are compatible with any personal 
computer that supports Mathcad 3.1 for Windows. However, many of 
the programs require a numerical solution of highly complicated 
integral equations that are computationally intensive. Where 
necessary to reduce processing time, the applications use 
trapezoidal approximations as an alternative numerical procedure 
to evaluate required integrals. Nevertheless, several programs 
still require several days to complete calculations on a 33 MHz, 

113 



386 personal computer. 

As previously mentioned, several of the antennas included in 
this project are not adequately reviewed by current texts. Thus, 
equations from many professional journals and doctoral 
dissertations are used for a number of the applications. 
Unfortunately, many errors existed in these sources, and 
resolution of these mistakes significantly slowed the progress of 
our research. 

The most disappointing aspect of the Mathcad application 
software aside from the excessive length of time required to 
analyze some types of antennas is the graphical presentation of 
the far-field radiation patterns. Mathcad's limited graphics 
flexibility, particularly in regards to three-dimensional and 
spherical coordinate plots, precluded better presentation of 
output data. 

The equations and data of this antenna analysis package 
could easily be transferred to another mathematics program, such 
as MATLAB. Although a MATLAB program may not be as easy for a 
new user to employ, it might offer advantages with respect to 
lower processing time and improved graphical output. 



114 



THE HELICAL ANTENNA 
MATHCAD SOFTWARE-HELIX. MCD 

When built to the proper specifications, the helical antenna possesses 
many qualities which make it suitable for a wide variety of 
communications applications. If the following conditions are satisfied 
the helix will exhibit a highly directional axial main lobe, low side 
lobe level, negligible mutual interference with adjacent antennas, low 
voltage standing wave ratio (VSWR), and resistive input impedance over a 
wide frequency band: 

.8 < Cj < 1 . 15 
n > 3 
12 < a < 14 

(Note: ). in a subscript indicates the dimension is in wavelengths. 
Mathcad equations can not use symbolic subscripts. Therefore, the symbol 
). will immediately follow the parameter in equations (i.e., O.) ). 

The helical antenna Mathcad application will compute the following 
parameters (Items with * indicate parameters that are calculated for 
both axial and peripheral feed geometries) : 

C = circumference of helix 
>. = wavelength 
a = pitch angle 
D Q = directivity 

p = relative phase velocity 

Eg j = Electric Field Components 

H' - Array Factor Phase Shift 
U = Radiation Intensity 
P rac j = Radiated Power 

P. = Antenna Input Resistance* 

T = Voltage Reflection Coefficient* 

z r = Reflection Efficiency* 

h em = Maximum Effective Height* 

G = Gain* 

EIRP = Effective Isotropic Radiated Power 

A em = Maximum Effective Aperture* 

AP = Axial Ratio 

PLF = Polarization Loss Factor 

BW = Bandwidth 

^high ~ Upper Frequency Limit 

^low = Lower Frequency Limit 

Acceptable Conductor Diameter 

Ex,y,z = Electric Field Cartesian Components 

8p,(}>p = Unit Polarization Vector Coordinate Angles 

o a = Antenna Unit Polarization Vector 
rmin = Minimum Distance to the Far-Field 



115 



The following data must be input based on known or estimated data 



D = 

S = 

L = 

n = 

d = 

f - 

m = 

lo 

x,y, 
i = 

Zo = 



Diameter of Helix (Center to Center) 
Spacing Between Turns (Center to Center) 
Length Along Conductor of One Turn 
Number of Turns 
Diameter of Helical Conductor 
Frequency of Interest 
Desired Mode 
= Antenna Feed Current 

z = Antenna Unit Polarization Vector Cartesian Coordinates 
Number of Increments in Degrees for Far Field Radiation Pattern 
Characteristic Feed Impedance 

= Incoming Wave Electric Field Unit Vector 



Enter input data here 



ow .= 






& 



(dimensionless ) 



D:=.1074 (meters 



L := .766 (meters 



n-=10 



Iturns 



f -=9.25 10° (Hz) 



m 



(dimensionless) 



Z :=150 (Q) 



S •= .0766 (meters 



i •= 360 (degrees) 



d:=.005 (meters 



Io := 1 (A) 



x-=l 



[meters 



z:=1000 (meters) 



> ' = 



(meters ) 



116 



Calculate helical geometric parameters and define constants 



c:=2.9979 10 (meters / sec) 



ti fi := 120-Ti (Q) 



C =n-D (meters 



X '•- - (meters / cycle) 

f 



C =0.33741 (meters 



Jl =0.3241 (meters / cycle) 



C/. : = — (dimensionless) 



S/. : = — (dimensionless) 



C/. = 1.04107 (dimensionless 



S/v =0.23635 (dimensionless) 



a-'=atanf— I (radians) 



La =— (dimensionless 



a = 0.22324 (radians) 



A - 18G 

ad a (degrees) 

71 



ad = 12.7908 (degrees 



Li. =2 36349 (dimensionless) 



Calculate helical antenna parameters: 



Define angular offset 6 
from helical axis: 



Minimum Distance to the Far-Field 



±-min-i 



2-71 

0, — ..2-7i (radians) 

i 



r =1.6X 



(mieters ) 



0, |- .. 2-7i ( radians ] 



r, :=5nS 



(meters) 



2(nSr , . 

r = meters 

2 



miin :=ma\(i) (meters) 



rmin = 3.83 (meters 



117 



Relative Phase Velocity p: 



La 



S/. + m + | — 
l2-iv 



j "i (dimensionless) 



p = 1.83736 (dimensionless) 



Array Factor Phase Shift \y_ 



»j'(6) =2-7i | Sacos(0) -| 

P, 



Electric Field Field Components E ^ , E , 



r- 



E(6):= 



sm fe) 


s,„h" ( ") 


sinf 


(t 2 )) 



cos(e) 



(V / IT.) 



EG 9) :=E(G) 



(V / m) 



E*(9) :=j -E(6) 



(V / m) 



Radiation Intensity U(0) 



U(8) := — ((ECO)!) 2 (W / solid angle) 



118 



Radiated Power P rar j : 



Prad := 



'2-71 





U(e)-sin(6)d0d(p (W) 



Prad- 1.2394 10 



(W) 



Directivity D ^ 



2 
Do ■= 12-CX -n-S/. (dimensionless ] 



Do2 .= 4 -» U ^ 
Prad 



(dimensionless ) 



Do = 30.73917 (dimensionless) Do2 = 26.89463 (dimensionless) 



Axial Ratio AR : 



AR ■: 



L/.- |sin(a) 

Pi 



(dimensionless ) 



AR =0.76309 



[dimensionless ) 



Effective Isotropic Radiated Power EIRP: 



EIRP '= Prad Do 



(W 



EIRP2 -=Prad-Do2 (W) 



EIRP =0.0381 



(W) 



EIRP2 =0.03333 



(W) 



119 



Polarization Loss Factor PLF: 



0p = atan| 



■Jx 4 V 



( radians ) 



<t»p = atan|- (radian: 



Op = 1.41421-10 * (radians) 



<J»p =0.7854 (radians) 



Ex :=E9fep)cos(ep) cos(<(>p)- E«9p)-sin(*p) 



(V/m) 



Ey := E6( 9p ) cos( 6p ) sin(. 4>p ) -»- E<j>^ Ojt ) ■ cos( 4»p ) 



(V/m) 



Ez :=Ee(9p)-sin(9p) - 1 



(V/m) 



oa .= 



J(|E\| ) 2 h- ( |Ey|) 2 + c|ez| r 



E\ 

Ey 
,Ez, 



(dimensionless ) 



oa 



0.5- 0.5j 
0.5 + 0.5j 

-9 99999 10 



-4 



(dimensionless) 



PLF 



:=U^v-oa|J 



(dimensionless ) 



PLF 



(dimensionless 



Radiation Resistance R 



r_i 



Ri := 



Prad 
(I Io|) : 



(fi) 



Rj =2.47881 10 



(fi) 



120 



Dual Parameters 
Axial Feed Peripheral Feed 

Input Resistance R: 



Ra:=14oJc/. (Q) Rp :=J-^i (Q) 



Ra = 1.42846" 10 2 (Q) Rp = 1.47012 10 2 (Q) 



Voltage Reflection Coef f ecient T: 



Ra 7 Rp " Z o 

o , rp : (dimensionless ) 



Ta ■ = (dimensionless) 

Ra-t-Z 



R P+ Z o 



(i 



fa =-0.02443 (dimensionless) Ip =-0.01006 (dimensionless 



Reflection Efficiency c, 



era := 1 - ( | Fa | ) (dimensionless: Q-p := ] _ ( |rp| ) (dimensionle 



era = 0.9994 (dimensionless) op =0.9999 (dimensionless 



121 



Gain G: 



Ga := era Do 



(dimensionless ) Gp := erp Do (dimensionless; 



Gadb :=101og(eraDo) (dB) 



Gpdb := lOlog(crpDo) (dB; 



Ga =30.72082 (dimensionless) Gp =30.73606 (dimensionless) 



Gadb = 14.87433 (dB; 



Gpdb = 14.87648 (dB) 



Ga2 — era Do2 (dimensionless) Gp2—apDo2 (dimensionless) 



Gadb2 := 10 log(era Do2) (dB; 



Gpdb2 : = 1 log( aj> Do2 )( dB ) 



Ga2 =26.87858 (dimensionless) Gp2 =2689191 (dimensionless) 



Gadb2 = 14.29406 (dB, 



Gpdb2 = 14.29622 (dB) 



Maximum Effective Aperture A, 



eiru. 



. era /. Do _,. _ 
Aema •= PLF 

4- a 



(m 2 ) 



Aemp:=fEiJ52.PLF (m 2 ) 

4-a 



Aema =0.25679 



m' 



Aemp =0.25691 



(m 2 ) 



. . . Era?. Do2 nT 
Aema2 •= PLF 

4 71 



(m 2 ) 



Aemp 2 •= — - PLF ( m^ ) 

4 71 



Aema2= 0.22467 



(m^ 



Aemp2= 0.22478 



(m 2 ) 



122 



Maximum Effective Height h pm : 



erru. 



hema =2 Rr 



Aema (m) 



hemp : 2 JRj emp (m) 



hema -2.5988 10 



m 



hemp = 2.59944* lO"-' ( 



m 



hema2 (Rr^^ (m) 

In 



hemp2 =2 JRr emp ~ (m) 



hema2 -2.43086 10 ' (m) 



hemp2 =2.43146 10 3 (m) 



Bandwidth : 



fhigh := 



1.15c 



(Hz) 



flow :=. 



(Hz 



fhigh = 1.02179- 10 (Hz; flow =7.10809- 10 (Hz 



BW = fhigh- flow (Hz 



BW =3.10979 10* (H: 



Acceptable Conductor Diameter 



dmin := .005 /. 



(m) 



dma\:=.05/. (m) 



dmin = 1.62049 \0 ' (m) 



dmax =0.0162 (m) 



123 



HELICAL ANTENNA FAR-FIELD RADIATION PATTERN 

For the purpose of this far-field radiation pattern, the helical antenna 
axis is equivalent to the Ex = grid line. The pattern is essentially 
symetric when rotated about the antenna's axis. 



Ex(6) .= E|8- -|cos(0) 



Ey(G) .= E|9 1 -sin(9) 



1 1 



097 



084 



0.71 



0.58 



Ey(&) 0.45 



0.32 



19 



006 



"0.07 



-0. 

























/ 


















/ 


















\ 

\ 






















\ 




\ 


















I 


\ 


\ 
















! 




\ 


\ 










/ 


t 






K 


\ 


\ 




y 




/• 


' 












^ 

n 


<£ 

F 


X 
















\. 


J 


"— O 









-0.25 -0.2 -0.15 -0.1 -0.05 "1.388 10 17 05 1 0.15 0.2 0.25 



E FIELD 



Ex(G) 



124 



THE BEVERAGE ANTENNA 
MATHCAD SOFTWARE-BEVERAGE. MCD 
The Beverage antenna is a single wire structure parallel to the ground. 
It is terminated with a load matching the characteristic impedance of 
the wire. Because there is little or no reflected energy from the 
antenna's termination, the Beverage antenna does not develop a 
significant standing wave. Therefore, it is known as a traveling wave 
antenna. The Beverage antenna is also known as a slow wave antenna 
since the relative phase velocity along the wire is usually less than 
one . 

The Beverage antenna is used for a wide range of frequencies, depending 
on its length and the characteristics of the ground under the antenna. 
It transmits and receives vertically polarized electromagnetic waves 
primarily through a cone shaped main beam pointing in the direction of 
the traveling wave. In the far-field, the electric field pattern above 
the ground can be considered rotationally symmetric with respect to the 
axis of the antenna. 

A Beverage antenna exhibits a highly directional main lcbe and resistive 
input impedance for frequencies corresponding to the following lengths: 

.5 < Lj < 2.0 (wavelengths) 

(Note: ). in a subscript indicates the dimension is in wavelengths. 
Mathcad equations can not use symbolic subscripts. Therefore, the symbol 

/. will immediately follow the parameter in equations (i.e., LJ.) to 
indicate the dimension is in wavelengths.) 

The Beverage antenna Mathcad application will compute the following 
parameters : 

L- = Length of Antenna in Wavelengths 

/. = Wavelength 
D Q = Directivity 

p = Relative Phase Velocity 

E = Electric Field (No Ground Effects) 

Et = Electric Field (Total Field Including Ground Effects) 

U = Radiation Intensity 

P ra q = Radiated Power 

Z Q = Antenna Characteristic Impedance 

T = Voltage Reflection Coefficient 

T v = Vertical Reflection Coefficient (Ground Reflection) 

e rv = Reflection Efficiency 

G = Gain 

EIRP = Effective Isotropic Radiated Power 

A em = Maximum Effective Aperture 

PLF = Polarization Loss Factor 
max = Angle of Maximum Radiation 
BW = Bandwidth 

fhiqh = Upper Frequency Limit 
fj_ ow = Lower Frequency Limit 

Acceptable Conductor Diameter 

125 



£r ' 



Ground Relative Complex Permittivity 



(Note: The subscript ' will be annotated as p in the application) 



"max 



Maximum Acceptable Conductor Diameter 



r mm = Minimum Distance to Far-Field 

R r = Radiation Resistance 

X = Electric Field Function 



The following data must be input based on known or estimated data: 



h = Height of Antenna above ground 

L = Length of Antenna 

d = Diameter of Conductor 

f = Frequency of Interest 

l = Number of Increments in Degrees for Far Field Radiation Pattern 

Zj_ = Load Impedance 

= Incident Wave Electric Field Unit Vector 

Incident Wave Arrival Angle 

Ground Conductivity 
Ground Relative Permittivity 

= Distance of Field Calculations 

= Input Current at Antenna Terminals 



u w 

e a 

= 

£ r = 

r ff 



Enter input data here: 



ow 



(dimensionless ) 



o -=01 



(mhos / m) 



d ■= .005 



(meters) 



(dimensionless , 



l :=360 



(degrees ) 



rff:= 1000 (meters) 



h=.l 



(meters) 



I Q - 1 (amps) 



Zl ■= 150 



(Q) 



L-=200 



(meters ) 



Ga :=— ^ — 20 

180 



(radians 



f:=16 10° (Hz: 



12 6 



Calculate Beverage antenna geometric parameters and define constants 



c: = 2 9979 10 (meters / sec; 



f 



(meters / cycle) 



oa 



[dimensionless ) 



>. = 1.87369- 10 (meters / cycle) 



ri := 1 20 7i (Q! 



U :=i 



(dimensionless 



1 9 

e. := 10 (Farads / mi) 

36ti 



L/. = 106741 (dimensionless) 



Calculate Eeverage antenna parameters: 

Define angular offset 
from Beverage antenna axis: 



Minimum. Distance to Far-Field 



* * 



radians i 



r :=1.6>. 



(m) 



0,|-| .71 

Li. 



( radians 



r = 5 L 



V = 



2-V 



(m) 



(m) 



Relative Phase Velocity p 



rmin =max(r) (mj 



rmin = 110" (m) 



p:=. 65891 



1000 



.0?852?821 



(dimensionless 



p =0.87551 



(dimensionless 



127 



Maximum length L m ^ v : 



Lma\ : = 



1 



I I — cos(Ga) 



(wavelengths ) 



Lma\ = 1.234: 



(wavelengths) 



Wavenumber k: 



k-ii 



(m" 



k =0.03353 



(m 



-1 



Electric Field Function X 



X(G) := — ( 1 - cos! ())) 

9 



(radians ) 



Electric Field Without Ground Effects E (0) 



E(0) 



30 kL I -sin(O) 



rll 



sin(X(G)) 



X(9) 



(YYtfar.) 



Relative Complex Dielectric Coefficient i r i 



crp := er- j 



2-Jl-f-E„ 



(dimensionless ) 



crp = 2- 1.125- 10 j 



(dimensionless ) 



126 



Vertical Reflection Coefficient fv: 



rv(e)-= 



erp cos| — 



- krp- sin — 6 



(dimensionless ) 



op COS] 



np- sm| — 



Electric Field With Ground Effects Et (6) 



Et(6) .= E(6)-U- rv(6) e 



-j -2-k-h cos 



(H) 



(V/m) 



Angle of Maximum Radiation 6^^ y 



Gma\ := aco: 



371 



( radians ) 



Gma\ =0.86001 



radians 



1 8() 
Gmax — =49.27486 



[degrees ) 



Radiation Intensity U 



U(6)-= — .(|E(e)|) 2 

2 % 



(W / solid angle) 



Umax '=U(0max) 



(W / solid angle) 



Umax = 19.17134 



(W / solid angle) 



129 



Radiated Power P ra( -j : 



Prad = 



U(G)-sin(0)dQd<!' 



(W 



Prad =32.66448 



(Wl 



Directivity D n : 



D„ := 



4 7i Umax 
Prad 



[dimension! ess 



D Q =7.37542 



(dimension! ess 



Polarization Loss Factor PLF: 



IX 2 



PLF 



U<T\v-oa|J 



(dimension! ess) 



PLF = 1 



(dimension! ess ) 



Characteristic ImDedance 



g|^ 



(Q) 



Z = 2.62626 lCr 



(Q) 



130 



Voltage Reflection Coef f ecient j: 



zi-z 

T: = (dimensionless) 

zuz 



T =-0.27295 (dimensionless; 



Reflection Efficiency e rw : 



erv:=l-(|r|) (dimensionless) 



erv =0.9255 (dimensionless) 



Effective Isotropic Radiated Power EIRP: 



EIRP:=PradD (W) 



EIRP =2.40914 10 2 (W) 



Gain G: 



G:=ervD (dimensionless) 



Gdb := 10 log ({tv D^ (dB) 



G =6.82594 



(dimensionless! 



Gdb = 8.34162 (dB) 



131 



Maximum Effective Aperture A prn : 



erv^D 

Aem: = - PLF (m 2 ) 

4 71 



Aem = 1.90698 10 4 (m 2 ) 



Radiation Resistance R r 



D . 2-Prad (Q) 

Rr = 



(I'ol) 



Rr =65.32897 



Bandwidth: 



(Q) 



2c 
fhigh := — (Hz) 

U 



fhigh =5.61713 10* (Hz) 



5c 
flow := (Hz) 

U 



flow = 1.40428- 10 8 (Hz) 



BW:= fhigh- flow (Hz) 



BW= 4.21285- 10 (Hz; 



132 



Maximum Effective Height h p, m : 



[RrAem 
hem =2 J — (m) 



hem = 1.14972 10 2 (m) 



Acceptable Conductor Diameter: 



dmax:=.01D. (m) 



dmax= 0.01067 (m) 



133 



BEVERAGE ANTENNA FAR-FIELD RADIATION PATTERN 

For the purpose of this far-field radiation pattern, the Beverage 
antenna axis is equivalent to the Ey = grid line (Note: Antenna height 
above ground is negible in the far-field) . The pattern is essentially 
symetric when rotated about the antenna's axis above the ground plane. 

Electric Field With Ground Effects 



Ex(8) -=Et(e)cos(e) 



Ey(9)-=Et(8)sin(e) 



0.018 



0.016 



0.014 



0.013 



0.011 



Ey(8)0.009 



0.007 



0.005 



0.004 



0.002 



III 



-0.005 "0.002 0.001 0.004 0.007 0.01 0.013 0.016 0.019 0.022 0.025 

Ex(6) 

134 



E FIELD 



Electric Field Without Ground Effects 



Ex(6) :=E(6)cos(0) 



Ey(6):=E(0)sin(0) 



o.i 



0.09 



0.08 



0.07 



0.06 



Ey(6)0.05 



0.04 



0.03 



0.02 



0.01 



















\ 




















\ 






















i 






























































r 


\ 


















\ 


\ 


















\ 


| 








/ 










\ 


\J 



















-0.02 "0.008 0.004 0.016 0.028 0.04 

Ex(6) 
— E FIELD 



0.052 0.064 0.076 0.088 0.1 



135 



THE SMALL LOOP ANTENNA 
MATHCAD SOFTWARE-SMLOOP.MCD 

The small loop antenna is a coil of one or more turns whose radius 
(a) satisfies the following: 

a < X/6n (m) 

where ().) is the wavelength of the frequency of interest. Small loops 
are commonly used to receive signals in the lower frequency regions. 
They are also used for direction finding and UHF transmissions. The 
efficiency of a transmitting loop antenna is typically very low. 
However, antenna efficiency can be improved by inserting a ferrite core 
in the loop, increasing loop perimeter, or increasing the number of 
turns . 

The small loop Mathcad applications will analyze three geometries: free 
space, horizontal loop, and vertical loop. Each geometry will also 
examine air and ferrite cores. Several antenna parameters, particularly 

those for loops in free space, can be calculated using more than one 
formula. Where this occurs, multiple results will be computed for 
comparison. Computations which are identical for all geometries will not 
be repeated. Computations with ferrite cores will use effective 
permeability (ji e )in lieu of free space permeability (\i Q ) . 

(Note: X in a subscript indicates the dimension is in wavelengths. 
Mathcad equations can not use symbolic subscripts. Therefore, the symbol 

). will immediately follow the parameter in equations (i.e., LX) to 
indicate the dimension is in wavelengths.) 

The small loop antenna Mathcad application will compute the following 
parameters : 

S = Cross-Sectional Area 

C = Circumference 

k = Wavenumber 

I = Wavelength 

D Q = Directivity 

E = Electric Field (No Ground Effects) 

E t = Electric Field (Total Field Including Ground Effects) 

U = Radiation Intensity 

U max = Maximum Radiation Intensity 

P rac j = Radiated Power 

T = Voltage Reflection Coefficient 

T v = Vertical Reflection Coefficient (Ground Reflection) 

1"^ = Horizontal Reflection Coefficient (Ground Reflection) 

e rv = Reflection Efficiency 

G = Gain 

EIRP = Effective Isotropic Radiated Power 

A em = Maximum Effective Aperture 

PLF = Polarization Loss Factor 
Bandwidth 

e r i = Ground Relative Complex Permittivity 
(Note: The subscript ' will be annotated as p in the application) 

136 



r min = Mi n i mum Distance to Far-Field 
R ohmic = 0nm i c Resistance 
R r = Radiation Resistance 

^s = Surface Impedance of Conductor 

e cd = Conduction-Dielectric Efficiency 

H e = Effective Ferrite Core Permeability 

X^ = Input Reactance 

Z^ = Input Impedance 

h em = Maximum Effective Height 

CR = Core Length to Diameter Ratio 

The following must be input based on known or estimated data: 

h = Height of Antenna above ground 

a = Radius of Antenna 

b = Radius of Conductor 

N = Number of Turns 

o c = Conductivity of Loop 

Rp/R = RR = Ohmic Resistance from Proximity to Ohmic Skin Effect Ratio 

D demag = Demagnetization Factor 

Hf = Permeability of Ferrite Core 

f = Frequency of Interest 

i = Number of Increments in Degrees for Far-Field Radiation Pattern 

o w = Incident Wave Electric Field Unit Vector 

o = Ground Conductivity 

£ r = Ground Relative Permittivity 

rff = Distance of Field Calculations 

I = Input Current at Antenna Terminals 

q = Loop Spacing 
cl = Core Length 
cd = Core Diameter 



137 



cw .= 



Enter input data here: 

'l\ 

nl /_,• , ■-<: m7 (mhos / m) 

(dimensionless) o- = 610 



b := .01 (meters) 



Ddemag := 4- 10"' (dimensionless 



er = 1 (dimensionless; 



i := 360 (degrees) rff:= 1 10 5 (meters) 



h:=2.5 (meters) RR:=.15 (dimensionless) 



oc := 5.8 10 (mhos / m) cl := 1 (meters) 



q:=03 (meters) f := 3 10 7 (Hz) 



— — = 1.5 (dimensionless) cd := .05 (meters) 

2b 



nf: = 4000-(4-ji-10" 7 ) (Henrys / m) Z o = 50 (Q) 



3 , , . I n := 1 (amps 



o 



N:=6 (dimensionless) a:=15 (meters; 



138 



c =2.9979 10 (meters / sec) 



% =120 7, 



(Q) 



V.= < 



(meters / cycle; 



36-7i 



(Farads / m) 



X = 9.993 (meters / cycle) 



H :=4-ti ia 7 (H / m) 



S:=7ia 2 (m 2 ) 



D:=2a 



(meters) 



S =0.07069 (m 2 ) 



D=0.3 



(meters) 



-^ =053015 

671 



(meters ) 



CR:= 



d 

cd 



(dimensionless ) 



C := 2 ■ 7i a 



(meters ] 



C =0.94248 (meters) 



oa 



V 





(dimensionless ) 



Calculate small loop antenna parameters in free space (air core) 



Define angular offset 
from small loop axis: 



Distance to Far-Field rmin: 



6:=oA*..2-7i 



(radians ) 



r Q :=1.6X (m) 



V =5D 



(m) 



$:=0,| — 1..2 7t (radians: 



r 2=" 



2-D" 



(m) 



rmin :=max(r) (m) 



rmin = 15.9888 (m) 



Wavenumber k: 





(m _1 ) 


k = 0.62876 


(m" 1 ) 



139 



Electric Field Without Ground Effects E (0) 



k-f-S-n -I -since) 

E(6)- = — e (J ' 

2rff 



(V/m) 



Radiation Intensity U( 6 ): 



rff 2 

U(9):=— (|E(e)|) 2 

2-nn 



(W / solid angle) 



Umax ~ Ul- 



(W / solid angle] 



Umax =9.30838 10 



-A 



(W / solid angle! 



Radiated Power P r3f j: 



Prad-= % f 2 (ka/(|l |) 



(w: 



Prad= 7.80909 10~ J 



(w; 



Prad2 := 



2 71 





U(8)-sin(e)d8cty 



(W) 



Prad2 =7.79817 10 



-3 



(W] 



Directivity D n : 



Do- =1.5 



(dimensionless) 



Do2 ■= 



4 Ti Umax 
Prad2 



(dimensionless ) 



Uo2 = 1.5 



(dimensionless) 



140 



Radiation Resistance (Rr; 



Rr :=20ti 2 |— 1 N 2 



(fi) 



Rr = 0.56225 



(Q) 



RrSrzi^-N 2 



(W 



(Q) 



Rr2 =0.56147 



(Q) 



Surface Impedance of Conductor R s 



Rs = 



h-Hr 



OC 



(Q) 



Rs = 1.42898 10 



-3 



(Q) 



Ohmic Resistance R 



ohmici. 



Rohmic = NaRs (RR4- 1) 



Rohmic =0.1479 



Conduction-Dielectric Ef f iciency je r r ] ) : 



ecd =■ 



Rr 



Rohmic ■+ Rr 



(dimensionless; 



ecd =0.79174 



(dimensionless [ 



Input Reactance (X -j ) : 



Xin :=2-7tf a n Q |ln|8 -I - 1.75 



(Q) 



Xin = 1.07924- 1(T 



(Q) 



141 



Input Resistance (R j ) : 

Rin: = Rr+Rohmic (Q) 

Rin =0.71015 (Q) 

Input Impedance (Z n ): 
Zi:=Rin+Xinj (Q) 



Zi =0.71015 +1.07924 10 2 j (Q) 



Voltage Reflection Coef f ecient j: 



Zi-Z 

T: = (dimensionless ) 

zuz 



r = 0.64337 +0.75901J (dimensionless) 

Reflection Efficiency e r ^: 
erv -= 1 _ ( I r| ) (dimensionless) 

en.- =9.98876 lO""" (dimensionless) 

Gain G: 



G:=GrvocdDo (dimensionless) 

Gdb := 101og(erv ecd Do) (dB) 

G =0.01 186 (dimensionless) 

Gdb =-19.25817 (dB) 



142 



Maximum Effective Aperture (A p 



Aem .= 

8 n 



(m 2 ) 



Aem = 11.91992 



(m< 



Aem2 := 



erv scd X Do 
4u 



(m 2 ) 



Aem2 = 0.09427 



(m 2 ) 



Effective Isotropic Radiated Power (EIRP) 



EIRP:=PradDo 



(W) 



EIRP =0.01171 



(W) 



EIRP2 :=Prad2Do 



(W) 



EIRP2 =0.0117 



(W) 



Maximum Effective Height (h PTn ) : 



hem := r Aem 2 



(m) 



hem = 0.26667 



(m) 



hem2:= r Aem2 2 



(m) 



hem2 =0.02371 



(m) 



Bandwidth: 



Polarization Loss Factor PLF: 



Bandwidth 



6 n a 



:hz 



PLF 



:= \\ow oa| ) 



(dimensionless) 



Bandwidth = 1.06029 10 



(Hz) 



PLF = 1 



(dimensionless) 



143 



SMALL LOOP ANTENNA FAR FIELD PATTERN IN FREE SPACE WITH AIR CORE 

For the purpose of this far-field radiation pattern, the small loop 
antenna axis is equivalent to the Ey = grid line. The pattern is 
symetric when rotated in the ($) dirction about the antenna's axis. 

Electric Field Without Ground Effects 



Ex(0) = |E(9)| cos(e) 



Ey(6) =|E(9)|sin(e) 



1 10 



8 10 



6 10 



"5 



4 10 



2 10 



-6 



Ey(6)-8.47 10 



-22 











































/ 


















) 


\ 
















J 


1 










































/ 
















\ 


\ 


\ 


















) 











































-2 10 6 



-4 10 -6 



-6 10 6 



-8 10~ 6 



-5 10 L 4 10~ 6 "3 10 6 -2 10 6 -1 10~ 6 -4.235 lo" 22 1 10~ 6 2 10 6 3 10~ 6 4 10 6 5 10 6 



— E FIELD 



Ex(6) 



144 



Calculate small loop antenna parameters in free space (ferrite core) : 



Effective Permeability {^ ) : 



jie-=. 



Hf 



1 + Ddemag ([if- 1) 



(Henrys/m) 



fie= 5.04663- 10 



(Henrys/m) 



Electric Field Without Ground Effects E (8) : 



E(6):=. 



kfS-jiel sin(8) 



v ' -(j -k-iff) 



2-rfif 



(V/m) 



Radiation Intensity U( 8) 



U(8):= — (|E(6)|) 2 

2 Tln 



(W / solid angle; 



Umax :=U|- 
12, 



(W / solid angle; 



Umax = 1.50127 10* 



(W / solid angle) 



Radiated Power P r ^: 



Prad: = 



2-ji 



U(8)sin(8)ded4 



(Wj 



Prad = 1.2577-10' 



(W) 



Directivity D ^ : 



Do- =1.5 



(dimensionless) 



Do2:= 



4 7i Umax 
Prad 



(dimensionless) 



Do2=1.5 



(dimensionless) 



145 



Radiation Resistance (Rr) 



Rr=20-« 2 -HVfej a (Q) 



Rr = 9.0681 1-10 6 (Q) 



Rrtr-l^l.N 2 (Q) 



(W) 



Rr2 =9.05542 10 6 (Q) 



Surface Impedance of Conductor R ^: 



Rs: JnHe (Q) 




Rs= 0.09056 (Q) 



Ohmic Resistance R n hmi c_L 



N-a-Rs 
Rohmic = (RR4-1) (Q) 



Rohmic =9.37266 (Q) 

Conduction-Dielectric Ef f iciency _(c r r j) : 

Rr (dimensionless) 

ecd := 



Rohmic ■+ Rr 
ecd = l 

Input Reactance (X -j ) : 



8-1- 1.7; 



Xin =4.3342 10 5 (Q) 



146 



(dimensionless) 



XriUUL RCilSUdULC |I\, ) 



li_l 



Rin:=Rr+Rohmic (Q) 



Rin =9.06812 10 6 (Q) 



Input Impedance (Z -j ) : 

Zi:=Rin+Xin-j (Q) 



Zi =9.06812 10 6 +4.3342 10 5 j (Q) 



Voltage Reflection Coef f ecient j: 

Zi-Z 
T : = (dimensionless) 

zuz 



r = 0.99999 +5.2587 10 _7 j (dimensionless; 



Reflection Efficiency c rw : 
ETV := 1 - ( |r| ) (dimensionless) 



etv =2.20048* 10" 5 (dimensionless; 



Gain G: 



G:=o"vecdDo (dimensionless) 



Gdb = lOlog(ervecdDo) (dB; 



G = 3.30071 10~ 5 (dimensionless) 



Gdb =-44.81392 (dB) 

Maximum Effective Aperture (A prri ) : 



. . erv ecd -X Do , o, 

Aem := (m^ 

4-71 



Aem =2.62295 10 4 (m 2 ) 

147 



Effective Isotropic Radiated Power (EIRP) 



EIRP:=PradDo 



(W) 



EIRP = 1.88655 10" 



(W) 



Bandwidth BW: 



Bandwidth := 



6 7i a 



(Hz) 



Bandwidth = 1 .06029 10 



(Hz) 



Maximum Effective Height (h pm ) 



hem-= J- Aem 2 



(m) 



hem = 5.02363 



(m) 



Polarization Loss Factor PLF: 



|\ 2 



PLF 



:= \|(jw oa|J 



(dimensionless) 



PLF = 1 



(dimensionless) 



148 



SMALL LOOP ANTENNA FAR FIELD PATTERN IN FREE SPACE WITH FERRITE CORE 

For the purpose of this far-field radiation pattern, the small loop 
antenna axis is equivalent to the Ey = grid line. The pattern is 
symetric when rotated in the ($) dirction about the antenna's axis. 

Electric Field Without Ground Effects 



Ex(0):=|E(8)| cos(6) 



Ey(6) :« |E(e)| -sin(G) 



0.035 



0.028 



0.021 



0.014 



0.007 



Ey(6)-5.204 10 



,-H 



"0.007 



-0.014 



-0.021 



"0.028 



-0.035 





> 














K 




/ 


/ 
















\ 


/ 


















j 


\ 


\ 














/ 


! 




\ 














/ 




/ 


















\ 


f 

\ 


















/ 


\ 


















^ 




^> 














r 





"0.02 -0.016 -0.012 -0.008 "0.004 0.004 0.008 0.012 0.016 0.02 

Ex(6) 

— EFELD 



149 



Calculate small horizontal loop antenna parameters over ground (air 
core) : 



Define angular offset 6 
from small loop axis: 



71 71 71 71 

2 2 i"2 



(radians) 



4:=0,|— | .2 7t 



(radians) 



Relative Complex Permittivity ( c r i ) : 



erp :=er- j - 



2-n-f-i 



(dimensionless) 



10 



erp =-3.6- 10' j 



(dimensionless ) 



Horizontal Reflection Coefficient (fh) 



m:=0..- 
2 



(increments) 



rh(6) 



i- ftjerp- 



cos(0)- Jerp- sin(0) 



^crp — 



cos(9) + Jap- sin(9) 



(dimensionless) 



cos 



fhl :=■ 



m 7i 



i / i 



. rm-71 
erp- sin 

i 



mri . /m-7t 

cos[ — I ■+ Jcrp- sin — 



(dimensionless ) 



Total Electric Field (Ethor; 



Ethor(G) := 



k-Sf-n -I n -an(6)-c" j *' 



o o 



2-rflF 



(.- 



rh(6)e 



j 2lchcos(9)} 



(V/m) 



Elhorl 



\ 2^ff / 



-j -2 k h 



cos(m i)l 



1 - Fhl e 



(V/m) 



150 



Radiation Intensity U( 8 ): 



rff 2 

U(6):=i=-(|Ethor(e)|r 

2 \ 



(W / solid angle) 



rff 2 2 

Umax := ( |max(Ethorl )| ) 

2 \ 



(W / solid angle) 



Umax =3.7199 10 



-? 



(W / solid angle; 



Radiated Power P ra( -j : 



f2-n 



Prad := 



-0 



|U(8)-sin(G)| dBd* 



(w; 



Prad =0.01016 



(W) 



Directivity D 



oj. 



Do:> 



4ji Umax 
Prad 



(dimensionless ) 



Do =4.59936 



[dimensionless ) 



Polarization Loss Factor (PLF! 



is 2 



PLF 



— UowoalJ 



(dimensionless ) 



PLF = 1 



(dimensionless) 



Radiation Resistance (Rr) 



Rr:=i±I!l.N 2 



(I'd) 



(Q) 



Rr = 0.73177 



(Q) 



151 



Surface Impedance of Conductor R ^: 



Rs = F** «»' 



OC 



Rs = 1.42898- 10 3 (Q) 



Ohmic Resistance R nhmir : 



Rohmic := (RR-t-1) (Q) 



Rohmic =0.1479 (Q) 



Conduction-Dielectric Ef f iciency j£ r r j ) : 



Rr 
ecd = (dimensionless) 

Rohmic +■ Rr 



ecd =0.83187 (dimensionless) 

Input Reactance (X -, ) : 



Xin:=2 7ifau |ln|8-| - 1.75 



(Q) 



Xin = 1.07924- 10 2 (Q) 



Input Resistance (R -j ) : 

Rin :Rr4 Rohmic (fi) 

Rm =0.87967 (Q) 

Input Impedance (Z -j ) : 

Zi: = Rirn-Xin j (Q) 



Zi =0.87967 +1.07924- 10 2 j (Q) 



152 



Voltage Reflection Coef fecient j: 
Zi-Z„ 



r-=- 



o 



Zi+Z, 



T = 0.64261 +0.75809j 



Reflection Efficiency s rv j 



Erv:=l-(|r|) : 



(dimensionless) 



(dimensionless ) 



(dimensionless) 



erv =0.01236 

Gain G: 

G :=Ervecd Do 

Gdb := 10 -log(Erv ecd Do) 

G= 0.04728 
Gdb =-13.25293 



(dimensionless) 

(dimensionless) 
(dB) 

(dimensionless) 
(dB) 



Maximum Effective Aperture (A prn ) : 



Aem 



_ ervgxU Do(PLF) 
4-ji 



Aem =0.37574 



(m 2 ) 



(m 2 ) 



Effective Isotropic Radiated Power (EIRP) 



EIRP := Prad Do 



(w; 



EIRP =0.04675 



(W) 



Bandwidth: 



Bandwidth '= 



6-Jia 



(Hz) 



Bandwidth = 1.06029- 10* (Hz) 



Maximum Effective Height (iy m ) : 



/RrAem . . . 

hem = | 2 (m) 

\ 

hem =0.05401 (m) 



153 



SMALL HORIZONTAL LOOP ANTENNA FAR FIELD PATTERN ABOVE GROUND WITH AIR 
CORE 

For the purpose of this far-field radiation pattern, the small loop 
antenna axis is equivalent to the Ex = grid line. The pattern is 
symetric when rotated in the ($) dirction about the antenna's axis. 



Small Horizontal Loop with Air Core Electric 
Over Real Ground 



Ex(9):=|Ethor(e)|cos[e + - 



Ey(G) :=|Ethor(0)|-sin[e+- 



5.5 10 



4.95 10 



3.85'10 




Ey(9)2.75*10~ 6 



2.2 10 



1.65 10 



l.rio -6 



5.5' 10~ 7 



-2M0-?.6M0~ 5 -1.2 10 5 -8'10~ 6 -4 10 Cl -1.694'10 _21 4 10 6 8 10 6 1.2M0~ 5 1.6M0 ^kT* 

Ex(6) 

— EFELD 



154 



core) : 



Incremental Horizontal Reflection Coefficient (fhl) 



m =0..- 
2 



^increments ' 



Fhl := 



. fm-n 
►s[ — — - jap- sin — 



(mm . rmji\ 
1 ■+ jap- sinl — 



(dimensionless; 



Total Electric Field (Ethor) : 



Ethor(G) =| 



'k-S-fjie I sin(9)e" j k " 
~ 2rff 



■(.- 



rh(6)e- J "" 



j -2khcos(S 



(V/m) 



Ethor 1 = 

m 



k-S-fjie-I -sinf— ^ 


-j krfli 

e J 


\ 2-rff 


r 



.j .2.kh cos(m -'A 



1 - Fhl e 



(V/m) 



Radiation Intensity U( 6 ): 



_«2 

U(0): = — (|Ethor(6)|) : 
2 \ 



(W / solid angle! 



rfl 2 
Umax : = ( |max(Ethorl )| ) 2 

2 % 



(W / solid angle) 



Umax =5.9995 1(T 



(W / solid angle) 



Radiated Power P rar j : 



f2-n 



Prad = 



|U(e)sin(6)|d6d4> (W) 



Prad = 1.63918- 10 3 



(w; 



Directivity D ^: 
4 n Umax 



Do:=- 



Prad 



(dimensionless ) 



Do =4.59936 



(dimensionless; 
155 



Radiation Resistance (Rr) : 



(I'd) 2 



Rr = 1.18021 10 7 (Q) 



Surface Impedance of Conductor R «~: 

y— (Q) 

Rs.= fe^ 



oc 



(G) 



Rs = 0.09056 
Ohmic Resistance R n hmi ci. 



Rohmic-= NaRs (RR.|- 1) (Q) 



Rohmic = 9 37266 (G) 



Conduction-Dielectric Ef f iciency jc r ri ) : 



Rr 

ecd := (dimensionless ' 

Rohmic 4 Rr 



ecd = 1 (dimensionless 



Input Reactance (X j ) : 



[8- - 1.751 



Xin =4.3342 10 5 (G) 



Input Resistance (R ^ ) : 
Rin " = Rr+ Rohmic (G) 



Rin = 1.18021 10 7 (G) 



Voltage Reflection Coef f ecient j: 

z.-z 

r •= (dimensionless) 

z> + z 

r = 0.64261 +0.75809j (dimensionless) 

156 



Reflection Efficiency c rv : 



crv := 1 - ( |r| ) 2 



(dimensionless) 



etv =0.01236 



(dimensionless ) 



Gain G: 



G:=erv ecd Do 



Gdb := 10 log(arvecd Do) 



(dimensionless) 
(dB) 



G= 0.05684 



(dimensionless ) 



Gdb =-12.45349 



(dB) 



Maximum Effective Aperture (A prn ) : 



Aem : 



_Er vscdrDo(PLF) 

4 71 



(m 2 ) 



Aem =0.45168 



(m 2 ) 



Effective Isotropic Radiated Power (EIRP) 



EIRP:=PradDo 



(W) 



EIRP =7.5392 10 3 



(W) 



Bandwidth BW : 



BW = 



6 n a 
BW = 1.06029 10 8 



(Hz) 
(Hz) 



Maximum Effective Height (h pm ) 



Polarization Loss Factor (PLF) : 



[Rr Aem „ 
hem •= } — - — 2 



\\ 2 



(m) PLF 



:=yow • oa|y 



(dimensionless ) 



hem = 2.37827- 1(T 



(m) PLF = 1 



(dimensionless 



157 



SMALL HORIZONTAL LOOP ANTENNA FAR FIELD PATTERN ABOVE GROUND WITH 
FERRITE CORE 



For the purpose of this far-field radiation pattern, the small loop 
antenna axis is equivalent to the Ex = grid line. The pattern is 
symetric when rotated in the ($) dirction about the antenna's axis. 



Small Horizontal Loop with Ferrite Core 
Electric Over Real Ground 



Ex(0):=|Ethor(0)|cos[e + - 



Ey(8) :=|Ethor(0)| sin|9+- 



0.022 



0.02 



0.018 



0.015 



0.013 



Ey(6) 0.011 



0.009 



0.007 



0.004 



0.002 







n 










^\ 










i 










\ 




































1 




































































































1 































-0.08 -0.064 -0.048 -0.032 -0.016 

Ex(6) 

— E FIELD 



0.016 0.032 0.048 0.064 0.08 



158 



Define angular offset 
from small loop axis: 



9:=0,- .71 

i 



(radians; 



*:=0,-..n 
i 



(radians) 



Relative Complex Permittivity ( e r i ) : 



op := er- j 



2nfe„ 



(dimensionless) 



10 



op =-3.6 10 j 



(dimensionless \ 



Vertical Reflection Coefficient (fv) 



rv(6) := 



op COS | |- 



erp- sin|- 



op cos 



«i- 



4 |op- sin| — 



(dimensionless 



m: = 0..i 



(increments ] 



Tvl := 



. In m 7t 



, it m-Jrtl . ( 7t m 7t 

op cos| | — — I I + Jap- sin| — 



(dimensionless ) 



Total Electric Field (Etvert; 



/k S f n I sin(8) e" J M < 

Etvert(0) := 2J (l + 

\ 2rff / 



rv(9)e 



-j -2khcos( 8)1 



(V/m) 



Etverl := 

m 



kSf^sinl— | 

\ 2rff 



m 7t\ -j -k- 



-j -2kh cos 



UTvl e 



W 



(V/m) 



159 



Radiation Intensity U( 8 ): 
U(6):=— (|Etvert(0)|) 2 

Umax : = ( |max(Etverl )| ) 

2 \ 



Umax =3.72332 10 



-3 



(W / solid angle) 

(W / solid angle) 
(W / solid angle) 



Radiated Power P rai -j : 



Prad: = 



|U(9) sin(G)| dOcty 



(W) 



Prad =001016 



(W) 



Directivity D n : 



Do:: 



4- ji Umax 



Prad 
Do =4.60367 



(dimensionless) 



(dimensionless) 



Radiation Resistance (Rr' 



Rr:.l**N 2 



(I'ol) 



:Q) 



Rr=0.73176 



[Q) 



Surface Impedance of Conductor R <-: 



Rs:= 



f*f-fti 



oc 



(0) 



Rs = 1.42898- 10 



!Q) 



160 



Ohmic Resistance R hmj ci. 



Rohmic '= N a Rs (RR+ 1) 



(Q) 



Rohmic =0.1479 



(Q) 



Conduction-Dielectric Ef f iciency ]c r , 



ecd •=■ 



Rr 



Rohmic + Rr 



(dimensionless) 



ecd =0.83187 

Input Reactance (X ., ) : 



(dimensionless) 



Xin:=2 7ifan [ln[8-| - 1.75 



(Q) 



Xin = 1.07924 10' 



(Q) 



Input Resistance (R -; ) : 



Rin = Rr-f Rohmic 



(Q) 



Rm =0.87966 



(Q) 



Voltage Reflection Coef f ecient j: 



Zi-Z 



r.=. 







Zi+Z, 



(dimensionless ) 



T = 0.64261 +0.75809j 



(dimensionless ] 



Reflection Efficiency e rw : 



erv .= 1 - ( |r| Y 



(dimensionless ) 



erv =0.01236 



(dimensionless) 



161 



Gain G: 



G = erv ecd Do 



Gdb:= lOlog(o-v-ecdDo) 



(dimensionless) 
(dB) 



G =0.04733 



(dimensionless) 



Gdb =-13.24888 



(dB) 



Maximum Effective Aperture (A PTn ) : 



. ervecdX Do(PLF) 
Aem •= - - 

4-ji 



(m 2 ) 



Aem =0.37609 



(m 2 ) 



Effective Isotropic Radiated Power (EIRP) 



EIRP =PradDo 



(w; 



EIRP =0.04679 



(W) 



Bandwidth 



Bandwidth : = 



6 7i a 



[Hz) 



Bandwidth = 1 .06029 10 



(HZ) 



Maximum Effective Height (h prn ) : 



: r Aem 2 
^0 



(m) 



hem = 0.05404 
Polarization Loss Factor (PLF) 



(m) 



10 1 
1 
C, 1 



(dimensionless \ 



|\2 



PLF 



:= UawoalJ 



ow 



(dimensionless) 



(dimensionless) 



PLF = 



(dimensionless) 



162 



SMALL VERTICAL LOOP ANTENNA FAR FIELD PATTERN ABOVE GROUND WITH AIR CORE 

For the purpose of this far-field radiation pattern, the small loop 
antenna axis is equivalent to the Ey = grid line. The pattern is 
symetric when rotated in the ($) dirction about the antenna's axis. 



Small Vertical Loop with Air Core Electric 
Over Real Ground 



Ex(0):=|Etvert(e)|cos(0) 



Ey(0) :=|Etvert(e)|-sin(9) 



1.8*10 



1.62 10 



-5 



1.44 10 



-5 



1.26 10 



-5 



1.08 10 



~5 



Ey(0) 9 10 



-6 



7.2 10 



-6 



-6 



5.4 10 



3.6* 10~ 6 



1.8" 10~ 6 







































/ 








^ 










/ 


















/ 


' 
















\ 






















1 






\ 


















j 




















/ 


I 


















/ 








\ 






















\ 














-6 10r< 


i. 


10 _6 -3.6 


10~ 6 -2.4 


10~ 6 -1.2 


10 6 


1.2 


10~ 6 2.4 


10~ 6 3.6 


10~ 6 4.8 


10 


'% 10 6 



— E FIELD 



Ex(9) 



163 



Calculate small vertical loop antenna parameters over ground (ferrite 
core) : 

Vertical Reflection Coefficient (Tv) : 



op cos 



rv(0) 



!H- 



apcos||-- 



op- sin — 

2 



op- sin — 
\2 



(dimensionless \ 



rvi •= 



11 m 
op cosh 

2 i 



it m n 
op- sin| 

2 i 



n m 
op cos|| 



ii m it 
+ |op- sm| r 



(dimensionless ) 



Total Electric Field (Etvert) : 



Etvert(G) •= 



fk S f-jie I sin(9) e" j k 
"" 2-rff 



(l-,^v(e)e• J • 2 ■ k • ,1C06(e, ) 



(V/m) 



Etverl := 



. _ - . . (m n\ .j krfn 
kSf-nel -sinl — I e J 



2-rff 



• j -2-k-h-cos 



1 + Tvl e 



yi 



(V/m) 



Radiation Intensity U ( 6 ) : 



U(6): = — (|Etverl(e)|) 2 
2 \ 



(W / solid angle; 



Umax : = ( |max(Etverl )| ) 

2 \ 



(W / solid angle) 



Umax =6.00502 10* 
Radiated Power P ra H : 



(W / solid angle) 



Prad •= 



|U(8)sin(6)| d6d4) 



(W) 



Prad = 1.63918- 10 3 



164 



(w; 



Directivity D n : 



Do = 



4 7i Umax 
Prad 



(dimensionl ess ) 



Do =4.60361 



[dimensionl ess ! 



Radiation Resistance (Rr) 



Rr :.i^l.N 2 



(l'»l)' 



(G) 



Rr= 1.18021 10 



(G) 



Surface Impedance of Conductor R s : 



Rs= fc!> 



oc 



(Q) 



Rs= 0.09056 



<G) 



Ohmic Resistance R 



ohmic. 



Rohmic := Na Rs (RR+ 1) 



(G) 



Rohmic =9.37266 



(G) 



Conduction-Dielectric Ef f iciency lc rr j) 



ecd :=■ 



Rr 



Rohmic ■+ Rr 



(dimensionless 



ecd = l 



(dimensionless) 



Input Reactance (X -j ) : 



Xm =2nfa|ie|In|8-| - 1.75 



Xin =4.3342 10 



(Q) 
(G) 



Input Resistance (R -j ) : 



Rin := Rr-+ Rohmic 



Riii = 1.18021- 10' 



<G) 

(G) 



165 



Voltage Reflection Coef fecient j: 



Zi-Z„ 



r:= 



ZUZ 







T = 0.64261 +0.75809j 



(dimensionless) 



(dimensionless ) 



Reflection Efficiency e rv : 



erv.= l-(|r|) 2 



(dimensionless) 



o-v =0.01236 



(dimensionless] 



Gain G: 



G: = erv ecd Do 



(dimensionless) 



Gdb := 101og(«rvecdDo) 



(dB) 



G =0.05689 
Gdb =-12.44948 



(dimensionless ) 



!dB) 



Maximum Effective Aperture (A em j : 



Bandwidth: 



Ervscd). Do(PLF) 
Aem ■= 

4 n 



(m 2 ) 



Bandwidth 



6 7i a 



(Hz) 



Aem =0.4521 



(m 2 ) 



Bandwidth = 1.06029- 10 8 



(HZ) 



Effective Isotropic Radiated Power 
(EIRP) : 



EIRP:=PradDo 



(W) 



Maximum Effective Height (h p>m ) : 



hem-= r Aem 2 



(m) 



EIRP =7.54613 10 3 



(W) 



hem = 2.37936- 10 2 



(m) 



Polarization Loss Factor (PLF) : 



oa : = 



(dimensionless) 



|\2 



PLF 



:= \\<w oa|J 



(0\ 
ow := 1 
Vl 



[dimensionless; 



(dimensionless) 



PLF = 1 



(dimensionless) 



166 



CORE 

For the purpose of this far-field radiation pattern, the small loop 
antenna axis is equivalent to the Ey = grid line. The pattern is 
symetric when rotated in the ($) dirction about the antenna's axis. 



Small Vertical Loop with Ferrite Core 
Electric Over Real Ground 



Ex(6) :=|Etvert(9)|-cos(0) 



Ey(6) :=|Etvert(6)|-sin(8) 



0.07 



0.063 



0.056 



0.049 



0.042 



Ey( 6) 0.035 



0.028 



0.021 



0.014 



0.007 







/ 










\ 










/ 










\ 












































1 
























, 
















1 




\ 




















\ 






















\ 


\ 



































-0.025 "0.02 -0.015 "0.01 

— E FIELD 



A" H 



■0.005 1.735 10 " 0.005 
Ex(6) 



0.01 



0.015 0.02 0.025 



167 



THE LARGE LOOP ANTENNA 
MATHCAD SOFTWARE-LGL00P.MCD 

The large loop antenna is a coil of one or more turns whose radius (a) 
satisfies the following: 

a > X/6n (m) 

where (X) is the wavelength of the frequency of interest. Large loops 
are not commonly used and are considered impractical if the radius 
exceeds one wavelength. Like the small loop, the efficiency of a 
transmitting large loop can be low. However, antenna efficiency can be 
improved by inserting a ferrite core in the loop or increasing the 
number of turns. 

The large loop Mathcad applications will analyze two geometries: free 
space and the horizontal loop. Each application for a specific geometry 
will examine air and ferrite cores. Several antenna parameters, 
particularly those for loops in free space, can be calculated using more 

than one formula. Where this occurs, multiple results will be computed 
for comparison. Computations which are identical for all geometries will 
not be repeated. Computations with ferrite cores will use effective 
permeability (n e )in lieu of free space permeability (\i ) . 

(Note: X in a subscript indicates the dimension is in wavelengths. 
Mathcad equations cannot use symbolic subscripts. Therefore, the symbols 
will immediately follow the parameters in equations (i.e., L\) in lieu of 
subscripts . ) 

The large loop antenna Mathcad application will compute the following 
parameters : 

S = Cross-Sectional Area 

C = Circumference 

k = Wavenumber 

X = Wavelength 

D Q = Directivity 

E = Electric Field (No Ground Effects) 

E t = Electric Field (Total Field Including Ground Effects) 

U = Radiation Intensity 

Umax = Maximum Radiation Intensity 

p rad = Radiated Power 

T v = Vertical Reflection Coefficient (Ground Reflection) 

T^ = Horizontal Reflection Coefficient (Ground Reflection) 

G = Gain 

EIRP = Effective Isotropic Radiated Power 

A em = Maximum Effective Aperture 

Bandwidth 

E r < = Ground Relative Complex Permittivity 

(Note: The subscript ' will be annotated as p in the application) 
r min = Minimum Distance to Far-Field 
R ohmic = Ohmic Resistance 
R r = Radiation Resistance 

R s = Surface Impedance of Conductor 

168 



ecd = Conduction-Dielectric Efficiency 
H e = Effective Ferrite Core Permeability 



'em 



= Maximum Effective Height 



CR = Core Length to Diameter Ratio 

The following data must be input based on known or estimated data: 

h = Height of Antenna above ground 

a = Radius of Antenna 

b = Radius of Conductor 

N = Number of Turns 

o c = Conductivity of Loop 

Rp/R = RR = Ohmic Resistance from Proximity to Ohmic Skin Effect Ratio 

D demaq = Demagnetization Factor 

(if = Permeability of Ferrite Core 

f = Frequency of Interest 

i = Number of Increments in Degrees for Far Field Radiation Pattern 

o = Ground Conductivity 

e r = Ground Relative Permittivity 

rjf = Distance of Field Calculations 

I = Input Current at Antenna Terminals 

q = Loop Spacing 

cl = Core Length 

cd = Core Diameter 

Z Q = Characteristic Impedance 

Enter input data here: 



a. = .46 

b := .05 



(meters) 
(meters) 



er:= 1 



rff:=M0 3 



(dimensionless ; 



(meters) 



i •= 360 



(degrees ) 



RR.= .15 



(dimensionless ) 



h:=.72 
oc:=5.8 lO 1 
q •= .03 



(meters) 
(mhos / m) 
(meters) 



cl = 1 



(meters! 



Z Q :=50 (Q) 



Ddemag '= 410* (dimensionless) 



nf: = 4000-(4 7i 1 0" "0 (Henrys / 

o- = 610 (mhos / m) 



-5--0.3 

2b 



m 



(dimensionless) 



I := 1 (amps) 



f .= 3.26 10° 


(Hz) 


cd=.05 


(meters) 


N:=l 


(dimensionless ) 



169 



Calculate large loop antenna geometric parameters and define constants: 



c:=2.9979 10 (meters / sec) t^:=120-ji 



(Q) 



C 9 

X '■-- (meters / cycle) e := 10" 

f ° 36n 



(Farads / m) 



X =0.9196 (meters / cycle) n =4 n 10' 



-7 



(H / m) 



S:=nV 



(m 2 ) 



D:r2 a 



(meters) 



S= 0.66476 (m 2 ) 



D=0.92 



(meters! 



— =0.04879 (meters) 

6 71 



cd 



(dimensionless) 



C =2-7ia (meters) 



C =2.89027 (meters) 



Calculate large loop antenna parameters in free space (air core) 



Define angular offset 6 
from small loop axis: 



Distance to Far-Field r 



miru. 



• : 0, — .. 2 7i 



»| — I 



( radians; 



(radians) 



(m) 
(m) 

(m) 

rmin :=max(r) (m) 



r o 


:= 1.6-X 


r l 


:=5D 


r 2 


._2D 2 

1 



rmin =4.6 



(m) 



Wavenumber k: 



k-i-« 



(m _1 ) 



k =6.83251 



(m -1 ) 



170 



m =0..i 



(increments) 



E(0): Ji!ll^2. e -( J ^) J1(k . a , m( e)) 

\ 2-rff / 



(V/m) 



B,,g^^[^ 



(V/m) 



Radiation Intensity U ( 6 ) : 



_«2 

U(6):=— (|E(6)|) 2 



(W/solid angle) 



=— (Kl) 



2- 



% 



(W/solid angle) 



Umax :=ma\(Ul) 



(W/solid angle) 



Umax = 1.57377 1CT 



(W/solid angle) 



Radiated Power P ra H : 



Prad := 



f2u 





U(6)-sin(e)dedit> 



(W) 



Prad = 1.09949- Iff 5 



(W) 



171 



Directivity D ^,: 



Do :=. 682-1- 



(dimensionless) 



Do =2.1435 



(dimensionless) 



Do2 := 



4 7i Umax 
Prad 



(dimensionless ) 



Do2 = 1.79871 



(dimensionless ) 



Radiation Resistance (Rr) 



2 C 2 
Rr := 60-7i -N 2 



(Q) 



Rr= 1.861 18- 10"* 



(Q) 



Rr2- = 



2-Prad-N' 

(I'd) 2 



(Q) 



Rr2 =2.19898 10' 



(Q) 



Surface Impedance of Conductor R^ : 



Rs:= 



k f K 



oc 



(Q) 



Rs =4.71058 10 



-3 



(Q) 



172 



_UIUU-H__ 



Rohmic:=-^-^(RR+ 1) (Q) 



Rohmic = 0.04984 (Q) 



Conduction-Dielectric Ef f iciency _(c r r j) : 



Rr 

ecd — (dimensionless) 

Rohmic + Rr 



ecd =0.99997 (dimensionless; 



Gain G: 



G:=ecdDo (dimensionless) 



Gdb:= lOlog(ecdDo) (dB) 



G =2.14344 (dimensionless) 



Gdb = 3.31111 



IdB) 



Maximum Effective Aperture (A em ) : 

Aem =.0543 (X C) (m 2 ) 

Aem =0.14432 (m 2 ) 

2 

. - ecd X Do 9 

Aem2 .= (m z ) 

4 71 

Aem2 =0.14424 (m 2 ) 

173 



Effective Isotropic Radiated Power (EIRP) : 
EIRP: = PradDo (W) 



EIRP =2.35675 10 3 (W) 



Bandwidth : 



Bandwidth :=- f 1 1 (Hz) 



a H 6 



Bandwidth =6.17143 10 (Hz) 



Maximum Effective Height (h ^ m ) : 



. . [Rr Aem . , . 

hem •= J — 2 (m) 

J % 



hem = 1.68821 (m) 



/RrAem2 _ , , 

hem2 •= | 2 (m) 



hem2 = 1.68775 (m) 



174 



For the purpose of this far-field radiation pattern, the large loop 
antenna axis is equivalent to the Ey = grid line. The pattern is 
symetric when rotated in the ($) dirction about the antenna's axis. 

Electric Field Without Ground Effects 



Ex(0) := |E(0)| cos(0) 



Ey(e):=|E(e)|sm(e) 



0.003 



0.002 



0.002 



0.001 



0.001 



Ey( 6) 



-0.001 



-0.001 



"0.002 



-0.002 



-0.003 




-0.003 -0.002 -0.002 "0.001 "0.001 0.001 0.001 0.002 0.002 0.003 

Ex(9) 

— EFELD 



175 



Calculate large loop antenna parameters in free space (ferrite core) 
Effective Permeability (j t e j : 



jie = = (H/m) 



1 ■+ Ddemag (nf- 1) 



fie =5.04663 10~ 3 (H/m) 



Electric Field Without Ground Effects E (9) 



(2-ji-fa-ne-lA ,. .-. 
o e - (j krti) J1(kasin(0)) (v/m) 
2rff / 



E, m : = p^e< J -',lL, 1 nH| (V/m, 



Radiation Intensity U( 8 ): 



rff 2 
U(0):= (|E(6)|) (W / solid angle) 

2 % 



rff 2 

U1 m =— (Kl) 2 < w / solid an g le » 



2 



% 



Umax :=max(Ul) (W / solid angle) 



Umax =2.5382- 10 9 (W / solid angle) 



176 



Radiated Power P rar ) : 



Prad := 



•2n 





U(e)-sin(G)d9d* 



(W) 



Prad = 1.77327- 10 



10 



(W) 



Directivity D n : 



Do:=.682- 
X 



(dimensionless ) 



Do =2.1435 



(dimensionless; 



Do2 •= 



4 7i Umax 
Prad 



(dimensionless) 



Do2 = 1.79871 



(dimensionless) 



Radiation Resistance (Rr' 



R r . = ,2 6 0-£fe N 2 



(D) 



Rr = 3.00174- 10 



10 



(Q) 



Rr2.=l^iN 2 



(N) 



(Q) 



Rr2= 3.54654 10 



10 



(Q) 



177 



Surface Impedance of Conductor R ^: 



Rs: 



._ J'fpe 



oc 



(Q) 



Rs= 0.29852 



(Q) 



Ohmic Resistance R ohm-ir. : 



Rohmic := NaRs (RR + 1) 



(fi) 



Rohmic =3.15832 



(Q) 



Conduction- Pi electric Efficiency jc r( 



ecd :=• 



Rr 



Rohmic t Rr 



(dimensionless; 



ecd = l 



(dimensionless) 



Gain G: 



G — ecd Do 



(dxmensionless) 



Gdb = 10 log(ecd Do) 



(dBl 



G=2.1435 



(dimensionless) 



Gdb = 3.31 123 



(dB) 



178 



Maximum Effective Aperture (A prr| ) : 



2 



. ecd X Do , o, 

Aem •= (m^) 

4-71 



Aem =0.14425 (m 2 



Effective Isotropic Radiated Power (EIRP) 



EIRP:=PradDo (W) 



EIRP =3.801 10 10 (W) 



Bandwidth : 



Bandwidth :=-[ 1 1 (Hz) 



a \ 6 



Bandwidth =6.17143 10 (Hz) 



Maximum Effective Height (h e 



[Rr-Aem . 
hem = J — -2 (m) 



hem = 6.77808- 10 3 (m) 



179 



LARGE LOOP ANTENNA FAR FIELD PATTERN IN FREE SPACE WITH FERRITE CORE 

For the purpose of this far-field radiation pattern, the large loop 
antenna axis is equivalent to the Ey = grid line. The pattern is 
symetric when rotated in the ($) dirction about the antenna's axis. 

Electric Field Without Ground Effects 



Ex(G).= |E(9)|-cos(e) 



Ey(6) •= |E(0)| sin(9) 



Ey(6) 




12 15 



180 



Calculate large horizontal loop antenna parameters over ground (air 
core) : 

Define angular offset 8 
from small loop axis: 



71 71 71 71 

2' 2 i"2 



(radians) 



-(ll- 



(radians) 



Relative Complex Permittivity ( c r t ) : 



erp '•= cr— j 



2 Tlf £ 







(dimensionless 



erp = l -3.31288 10 j 



[dimensionless ) 



Horizontal Reflection Coefficient (Th) 



m =0..- 
2 



(increments 



._ cos(e)-^ap- sin(6)' 
cos(0)-K *>}zrp- sin(6)' 



(dimensionless) 



cos 



fhl :=■ 



m 7t 



. rm-ji 
|erp- sin 



fm-jj 
1 



(m m 
— 



(dimensionless) 



181 



Total Electric Field (Ethor) : 



Eth or(6).= i!^'M.-<i^ 

\ 2-rff 



-Jl(kasin(e))(l-rh(e)e- j - 2khcos(e) ) 



(V/m) 



Ethor 1 := 



l ^oV e -(j^).f J1 f k . a . sin fn L n 1 || 



2-rff 



■ j -2-k-h-coslm 



H 



\ i 



(V/m) 



Radiation Intensity U(8) 



_cr2 

U(9):=— (|Ethor(0)|) : 

2-nn 



(W/solid angle) 



Umax • = ■( |max(Ethoii )| ) 

2 \ 



(W/solid angle; 



Umax =7.03652 1CT 



(W/solid angle) 



Radiated Power P 



radl 



f2-7l 



Prad := 



|U(9)-sin(8)| dOcty 



(W) 



Prad = 1.13445- 1(T 



(W) 



Directivity D n : 



Do 



4 ji Umax 
Prad 



(dimensionless ) 



Do =7.79439 



(dimensionless) 



182 



Radiation Resistance (Rr; 



Rrsl^N 2 



(I'd) 



Rr = 2.2689- 10 J 



Bandwidth: 



(Q) 



Bandwidth :=- 1 1 

a \ 6 



(Hz) 



Bandwidth =6.17143 10° 



(Hz) 



Surface Impedance of Conductor R s : 



Rs:= 



Wh p 



oc 



(Q) 



Rs =4.71058-10 



>~3 



(Q) 



Ohmic Resistance R n hTn-j r. 



Rohmic := N a Rs (RR + 1) 



:Q) 



Rohmic =0.04984 



(fi) 



Conduction-Dielectric Ef f iciency _(£ r ^) 



ecd :=• 



Rr 



Rohmic •+ Rr 



(dimensionless) 



ecd =0.99998 



183 



(dimensionless) 



i-ia^-Luiu-iu lj l ico li vc ftpeiuuic 1-rt.pnv • 



Aem 



eed X Do 
4ji 



(m 2 ) 



Aem =0.52452 



(m 2 ) 



Gain G: 



G:=ecdDo 



(dimensionless) 



Gdb := 101og(ecd Do) 



IdB) 



G = 7.79422 



(dimensionless ) 



Gdb = 8.91773 



(dB) 



Maximum Effective Height (h prri ) : 



hem = r Aem 2 



(m) 



hem = 3.55347 



(m) 



Effective Isotropic Radiated Power (EIRP) 



EIRP := Prad Do 



(W) 



EIRP =8.84235 1CT 



(W) 



184 



CORE 



For the purpose of this far-field radiation pattern, the large loop 
antenna axis is equivalent to the Ex = grid line. The pattern is 
symetric when rotated in the ($) dirction about the antenna's axis. 



Large Horizontal Loop with Air Core Electric 
Over Real Ground 



Ex(6)-=|Ethor(8)|cos 9 + - 



Ey(8) •=|Ethor(e)|-sin[e+- 



0.005 



0.004 



0.004 



0.003 



0.003 



Ey(9)0.002 



0.002 



0.001 



0.001 



r 


\ 














/ 


"\ 




\ 














/ 
























\ 


















/ 


\ 


















/ 




































I 




















/ 




































\/ 






> 





-0.005-0.004 -0.003 "0.002 ~0.001 

Ex(6) 

— E FIELD 



0.001 0.002 0.003 0.004 0.005 



185 



Calculate large horizontal loop antenna parameters over ground (ferrite 
core) : 

Total Electric Field (Ethor) : 



Ethor(6): = 



'2-n-f-a ne I 



o] -(j k-iff) 



2-rff 



e -vj *■■"/. Ji(k- as in(e))Al- rh(9)e 



(i- 



-j -2khcos( 



.,) 



(V/m) 



Ethor 1 = 



* rf ^a^.f J x r j = *vLiM ■«""*■" 



2rfiF 



\ i 



H) 



(V/m) 



Radiation Intensity U ( 8) 



U(6):=— (|Ethor(0)|) 2 
2 % 



(W / solid angle) 



rfi 2 
Umax = ■( |max(Ethorl )| ) 

2 \ 



(W / solid angle; 



Umax = 1.13486 10 



10 



(W / solid angle) 



Radiated Power P rar ; : 



Prad := 



f 2 n 



• 



|U(e)-sin(8)| dBd4> 



(W) 



Prad = 1.82966 10 



10 



(W) 



186 



Directivity D n : 



4 -n- Umax (dimensionless 

Do := 

Prad 



Do =7.79439 



Radiation Resistance (Rr; 



(I'd) 



Surface Impedance of Conductor R c 



oc 



(dimensionless ] 



D . 2 -Prad KT 2 (Q) 

Rr •= N 



R]=3 6d931 10 



R S := liSS IQI 



R S = 0.29852 



Ohmic Resistance R nhnr j r 



Rohmic:= N aRs (RR+ 1) (Q) 



Rohmic =3.15832 (Q) 



187 



Conduct ion- Pi electric Ef f iciency ]| rrj ) 



Rr 

ecd := (dimensionless ) 

Rohmic + Rr 



ccd = 1 (dimensionless 



Gain G: 



G"=ecdDo (dimensionless) 



Gdb := lOlog(rcdDo) (dB) 



G = 7.79439 (dimensionless) 



Gdb -8 91782 (dB) 



Maximum Effective Aperture (A f 



ecd /. Do 9 

Aem "= (m z 

4 n 



Aem =0 52453 (m 2 ) 

Effective Isotropic Radiated Power (EIRP) : 
EIRP:=PradDo (W) 



EIRP = 1.42611 10 n (W) 



188 



Bandwidth 



Bandwidth :=-■ 1 1 I (Hz) 

a \ 6 n 



Bandwidth =6.17143- 10 (Hz 



Maximum Effective Height (h prri ) 



IRj Aem „ 
hem := \ 2 (m) 

\ 



hem = 1.42709* 10 4 (m) 



189 



LARGE HORIZONTAL LOOP ANTENNA FAR FIELD PATTERN ABOVE GROUND WITH 
FERRITE CORE 



For the purpose of this far-field radiation pattern, the large loop 
antenna axis is equivalent to the Ex = grid line. The pattern is 
symetric when rotated in the (<)>) dirction about the antenna's axis. 



Large Horizontal Loop with Ferrite Core 
Electric Over Real Ground 



Ex(6) •= |Ethor(e)|-cos|e+i 



Ey(6).= |Ethor(G)| sin Q+- 



Ey(6) 9 




8 12 16 20 



190 



THE BEDSPRING ANTENNA 
MATHCAD SOFTWARE-BEDSPRIN.MCD 

A bedspring (or curtain) antenna is a two dimensional array of identical 
horizontal dipoles. The vertical stacks of dipoles are referred to as 
bays. A bedspring antenna is built from two or more bays. Bedspring 
assemblies are normally designed for high frequency (3 - 30 MHz) 
operations. The beam maximum can be steered in either the azimuthal or 
vertical directions by adjusting the phase of the element feed currents. 
Bedspring antenna Mathcad applications assume that all elements lie in 
the y-z plane, that all element excitation currents have identical 
maximum amplitude, that there are only 2 dB of losses associated with 
the antenna, and that a perfect reflector screen is located in the -x 
half space. 

(Note: Mathcad equations cannot use symbolic subscripts. Therefore, 
symbols like X will immediately follow the parameter in equations in 
lieu of subscripts. ) 

The bedspring antenna Mathcad application will compute the following 
parameters : 

D = Maximum Physical Dimension of the Array 

k = Wavenumber 

/. = Wavelength 

D Q = Directivity 

Eq^ = Electric Field (0) Component for an Individual Bay 

Ea^ = Electric Field {$) Component for an Individual Bay 

S x = Array Factor for Perfect Image Reflector 

Sy = Array Factor for Multiple Bays 

A, B = Electric Field Coefficients 

Eg t = Electric Field (0) Component (Including Ground Effects) 

EA t = Electric Field (<j>) Component (Including Ground Effects) 

U = Radiation Intensity 

U max = Maximum Radiation Intensity 

P rac j = Radiated Power 

T v = Vertical Reflection Coefficient (Ground Reflection) 

r^ = Horizontal Reflection Coefficient (Ground Reflection) 

G = Gain 

EIRP = Effective Isotropic Radiated Power 

A em = Maximum Effective Aperture 

BW = Bandwidth 

E r i = Ground Relative Complex Permittivity 

(Note: The subscript ' will be annotated as p in the application) 
r min = Minimum Distance to Far-Field 
R r = Radiation Resistance 
n em = Maximum Effective Height 
|I I = Magnitude of Antenna Feed Current 
^x, y, z = Electric Field Spatial Components 
o a = Antenna Unit Polarization Vector 
^high = Upper Operating Frequency 
^low = Lower Operating Frequency 

191 



PLF = Polarization Loss Factor 

o a = Antenna Unit Polarization Vector 

6 p = Coaltitude (Deflection Angle from +z Axis) for Polarization Loss 

ij>p = Azimuth Angle for Polarization Loss 



The following data must be input based on known or estimated data 

M = Number of Elements per Bay 

N = Number of Bays 

Zj_ = Height of ith Element Above Ground 

Z^-Z-l.^ = Vertical Spacing of the ith Element 

Yj_ = Horizontal Position (Center of Dipole) of i th Bay 

Y 1 -Yj L _2 = Horizontal Spacing of the i t " Bay 
X]_ = Reflector Position 

1 = Half-Length of Each Element 
f = Frequency of Interest 

i = Number of Increments for Far Field Radiation Patterns 
(Note: Due to Mathcad restrictions on matrix size, i < 125) 
o = Ground Conductivity 
e r = Ground Relative Permittivity 

rff = Distance of Field Calculations 
I m = Input Current at Element Terminals 

Op = Coaltitude (Deflection Angle from + z Axis) for Azimuth Plot 
$q = Azimuth Angle for Elevation Plot 
C 1 = Relative Current Amplitude of i^" Element 
O = Vertical Scan Angle (Coaltitude) 
4> = Azimuthal Scan Angle 

(x,y,z) = Coordinates for Unit Polarization Vector 
o w = Incoming Wave Unit Polarization Vector 



192 



Enter input data here 



M:=4 



[elements) 



f := 10 10° (Hz 



N:=2 



(bays) 



er:=10 (dimensionless) 



i =90 



(increments ) 



rff = 1-10 (meters) 



Z := 



Y = 



55 


55 


42 


42 


29 


29 


16 


16 





26" 





26 





26 





26 



C:= 



ow = 



J_ 

_1_ 




(m) 



(m) 



(amps 



(dimensionless ) 



Im ~ 1 (amps ) 



o := .01 (mhos / m) 



Xl:=7 (m) 



1 = 11 (m) 



Qo:--l 



radians ) 



(f»o — (radians) 



x:=1000 (m) 



y =50 (m) 



z:=50 (m) 



193 



Calculate bedspnng antenna geometric parameters and define constants 



c =29979 10 (meters/sec) 



\ =120-7i 



(fi) 



).:■ 



(meters /cycle) 



- 1 irr 9 
36ti 



(Farads/m) 



). = 29.979 (meters/cycle) . 7 

H - = 4 n 10 (H/m) 



D =j( Y 0,N-l) 2 +( Z 0,o) 2 



(m) 



D =60 83584 



(m) 



Calculate bedspnng antenna parameter: 



Define angular offset 6 
from; y-z axis: 



Distance to Far-Field r 



mini 



oA - 

2 i 2 



{ radians ) 



r :=1.6X 



r,:=5.D 



(m) 



(m) 



71 / Tt Tfl 7t 



Wavenumber k: 



( radians ] 



2D' 



F 2 - 



(m) 



rmin :=max(r) (m) 



rmin =3.04179 10 (m) 



k-i! 



k = 0.20959 



(m -1 ) 



194 



Ground Relative Complex 
Permittivity g r i j_ 



J o 

erp = cr- 



2i\fc (dimensionless) 



erp = 10— 1 8 j (dimensionless; 



Vertical Reflection Coefficient £ v : 



,, l (increments 



/ — : — 2 

rv(e) , r °TCos(0;-^- i in(e)_ (dimensionless) 



wpcos(Q)i- ^arp- sin(O) 



crpcosl - erp- sin 



Ivl •= 3 __ (dimensionless 



npcos] 4 Jap- sin — | 



Horizontal Reflection Coefficient £>-, : 



cos(6)- Jerp- sin(9) 
In(fj).= ' (dimensionless 

cos(6) + ^erp- sin(6) 



wjn fw-m 

cosl 1 - [op- sin — 

fhl '= - — (dimensionless) 



cosj — | + [op- sin — 



V1- 



195 



Electric Field Coefficients A, B: 

(increments ) 
q = 1..M 



Ej kZ . „(cos(e)-cos(Goj) / -j -2-k-Z . cos(G)\ 

C q _ )0 e q -''° Al-rv(9)e «" 1 '° } 



(dimensionless ) 



'« "/ j C q-l.o' 



Al = > C . -e q 



/ M l( .\ I ■ t, 7 (**\\ <dimensionl( 

- i.o'Ht - cos(6o y -j 2kZ q _ 1,0 cos n~ j 



ess 



q 



Ej kZ n -(cos(Gj-cos(6o)) / -j 2 k Z . -cos(6)\ 

C q _ ]0 e q - K0 U4-rh(0)e q " 1 '° I 



(dimensionless ) 



Bl :=V C . -e 

« / i q- 1,0 

q 



j kZ _ , Ycosfc-ij - cos(Bo)J f -j -2-k-Z j -co S p_in I (dimensionless) 



Electric Field Components for Individual Bay EpmJ 



cn ,, ,,, ,,, T e* J cos(k lsin(G) -sin(^))- cos(k 1) . ,.. _ ..„. . 

E61 (0, 4>) :=-j 60 Im - — ^^ ■ — -sin((fi)cos(9)A(9) (V/m) 

111 1 - sin(9) 2 sin((^) 2 



v~0..i (increments; 



Hvt)1 



tr cosl k-lsinl — - 1 sinf — + — -] I - cosfk 1) 
e jkrff . 2 ill .in vn\ iw 



E611 :=-i -601m- S 1—-! i— -^ -sin— + — -cos -Al 

«ff . M 2 /» vt\ 2 I 2 . 

- sin| sin - — + — 

i / \ 2 i 

(V/m) 



,-,, „ ,s ^^ t e~ J ' cos(k-l-sin(0)-sin(6)) - cos(k-l) ,,* n ,** 

E(j)l(0,(t)) :=j 60 Im - — — - - cos($) B(G) (V/m) 

^ 1- sin(0) 2 sin($) 2 



-j krff °° s ^ 1 sm — -sin ~ + — - cos(kl) 

E*ll :=j 60 Im^ i LiJ 1 2 ill cos.- n + H iBl 

* , M 2 ./, v^ 2 I 2 ij 

1 - sin| j sin] 1- — | 



i / I I i 



Sy(M) ■= J] 
qi 



j k-Y -sin('J) (sin(<?)- sin( Oo ) ) 

e 



W,V " ^_| 



J kY o. q i-r sin l 



qi 



Array Factor for Ideal Reflector S v : 



Sx(e,4>) :=l-e- J - 2kXlsi * e > c ° s (*) 



-j 2k Xlsmtl co S (-l4-1 



(V/m) 



Array Factor for N bays S y : 



ql:=l..N (increments) 



(dimensionless ) 



(dimensionless ) 



(dimensionless) 



Sxl := 1 - e 

w,v (dimensionless) 



197 



Total Electric Field Components E g t a-j-J. 



Eet(6,4t) :=E61(e^) Sx(6,(j.)Sy(e,«))) 



(V/m) 



EOtl :=E011 Sxl Svl 

W . V W ' V W V W , V 



(V/m) 



E$t(G,$) -= E4>1(G,4.) Sx(B,4)) Sy(G,4>) 



(V/m) 



Editl :=E411 Sxl Svl 

' W . V W , V u , v • W , V 



(V/m) 



Radiation Intensity U(Q) 



1 

U(6,4):=— [(|Eet(0^)|) 2 +(|Eit(e^)| 



(W/solid angle) 



Ul 



rfT f 



(h 1 ...D 2 *(l«"w,.D : 



(W/solid angle) 



Umax :=ma\(Ul ) 



(W/solid angle) 



Umax = 8.8082- 10" 



(W/solid angle; 



Radiated Power P 





71 


' 71 




2 


2 


Prad := 




U(M) 




71 ' 





. 


2 





rad_L 



(W) 



Prad =9 79181 10 



198 



(W) 



Directivity D n : 



_ 4 n Umax (dimensionless ) 

Do = 

Prad 



Do = 1.13041 10 (dimensionless) 



Magnitude of Antenna Feed Current 



loi 



Io:=M-N-lm (amps) 



Io =8 (amps 



Radiation Resistance (R r \ 



Rrs-lSSi (Q» 



diolr 



Rr = 30.5994 (Q) 



Gain G: 



Gdb := lOlog(Do)- 2 



G =71.32377 



(dBl 



Gdb = 18.53234 (dB; 



Gdb 

G = 10 10 (dimensionless 



199 



(dimensionless) 



Polarization Loss Factor PLF: 



0p •= atan 



ff7~ 



radians ) 



4>p := atan I - 



(radians) 



8p = 1.5209 



( radians ) 



+p =0.04996 



( radians ) 



E\ :=Eet(0p,^p)-cos(0p)-cos(^p)- E<j>t ( Gp , <|>p ) sin(ij>p) 



(V/m) 



Ey :=Eet(ep,*p)-cos(9p)-sin(^p)- E<t>t(9p,<t>p)cos(<|>p) 



(V/m) 



Ez .= .Eet(ep.$p)-sinfGp; 



(V/m) 



oa := 



E\ 



E> 



-/(|Ex|) 2 +(|Ey|) 2 +(|Ez|) 2 \E 



(dimensionl ess) 



oa = 



-0.03302 + 0.03754j 
-0.66366 + 0.74634J 
-3.33112- 10~ ? - 4.40961 id" 



(dimensionless) 



PLF 



:=Uowoa|J 



(dimensionless) 



PLF =0.54991 



(dimensionless; 



Maximum Effective Aperture (A f 



Aem ' = 



X Do PLF 

4-71 



(m 2 ) 



Aem =4.44582 10" 



200 



Effective Isotropic Radiated Power (EIRP) 



EIRP =PradDo 



EIRP = 1.10687 10 5 



(W) 



(w; 



Maximum Effective Height (h pm ) : 



[Rr Aem 
hem .= J — 2 (m) 



hem = 37.99241 (m) 



Bandwidth BW : 



Half-Wave Assembly Symmetric Feed Assembly 



R2 =-£. 



41 (Hz) ihighl := 1.5-0.2 (Hz 



f/.2 =6.81341 10 6 (HZ) flowl = .98f7.2 (HZ 



fhigh := 1.02T/.2 (Hz) 



BW1 = fhigh 1- flow 



(H: 



flow := .98 0.2 (Hz) BW1 = 3.54297 10 6 



BW = fhigh- flow (Hz. 



BW= 2.72536 10 5 



201 



BEDSPRING ANTENNA AZIMUTHAL FAR FIELD PATTERN 

For the purpose of this far-field radiation pattern, the bedspring 
antenna lies on the Ey = grid line. Three field patterns are 
developed: the (8) component of electric field, the ($) component of 
electric field, the total electric field. The user must select the 
desired coaltitude (8 q ) for which the patterns will be graphed. 



Theta Component of Azimuthal Electric Field 



2 



E0tx(t) := |Eet(eg,*)|-cos|4n.- 



E6ty($):= |E8t(Qg,+)|-sinU + - 



4 10 
3.6 10 
3.2 10 
2.8 10* 



2.4 10 



r 19 

-19 

-19 

19 

-19 



E6iy(9) 2 10 
1.6 10 



1.2 10 
8 10 
4 10 



,"19 
,-.9 
,-19 
,-20 
,~20 





























j 


\ 


/ 


















\ 


1 












s~ 




l 


i \ 


j 






"\ 












\ 


i j 


/ 


^ 








\ 




N \ 


\ 


/ 


/ 




/ 






\ 




s 


\\ 


,< 


/ 




/ 








V 




V 


'// 




/ 










V 


^_ 


\ 


/ 


















\ 


/ 










-tf-ro"^ 


%'^U 


10 _i ^ 4 1 


\<r ll \.7' 


^ 


1.2-1 


_1 i4'l 


-1 ?.6" 


0~ 1 4 3 .8*1 


(r-Wr 11 ' 



E FIELD 



EGix($) 



Phi Component of Azimuthal Electric Field 



E<frtx($) := |E<H(6g,<fr)| cos|$ + - 



E<j)ty(<t)) :=|E$t(8g,$)| sin[$-i-- 



1.5M0 17 



Efry($) 7.5 10 
6 10 




4.5-10 18 

3*10 -1 



1.5 10 



-3 rt32.i*10 ^1.8 

— E FIELD 



10 '^1.2 10~ 12 "6 10~ Li 'l.926 - 10~% 4 10~ 19 l.ri0~ 18 1.8*10~ 18 2.4"103 1 } ! 18 
E4>lx( 4> ) 

202 



Total Azimuthal Electric Field 



Et(f) := J( |E9i(0g,*)| ) 2 + ( |E*t(6g,*)| 



Etx(^) •= |Et(^)|-cos|H- 



Ety(*).= |EtU)|-sin|H- 



1.6 10 



1.44 10 



1 28 10 



1.12 10 



9.6 10 



,-n 



Ety(*) 8 10 



6.4 10 



4.8 10 



3.2 10 



1.6 10 




-3* 10-2^ 10 ia -1.8'10 lii 1.2'10~ 1!L 6'10~ 12 T926 , 10^ 3 ^ 10 19 1.2*10 18 1 8* 10 18 2 .4* lO^o" 18 

EtA<<?) 

— E FIELD 



203 



BEDSPRING ANTENNA ELEVATION FAR FIELD PATTERN 



For the purpose of this far-field radiation pattern, the bedspring 
antenna lies on the Ex = grid line. Three field patterns are 
developed: the (0) component of electric field, the ($) component of 
electric field, the total electric field. The user must select the 



desired aximuth angle 
graphed. 



.) for which the elevation patterns will be 



Theta Component of Elevation Pattern of 
Electric Field 



E9ix(9) := 



EOtj — e,4>g 



*g =0 
cos(6) Eety(6) 



E9t| — e,$g 



sin(G) 



l 
0.8 

0.6 

0.4 

0.2 

Eftyce)-5.551"10~ 17 

-0.2 
-0.4 

-0.6 

-0.8 

-1 











1 















































































































































































-1 -0.8 -0.6 -0.4 -0.2 -5.551 10~ 17 0.2 0.4 0.6 0.8 1 

E9ix( 6) 

— E FIELD 



Phi Component of Elevation Pattern of 
Electric Field 



E$tx(8) := 



E(j)t|-- 9,<fcg' 



■cos(G) 



E*ty(8) := 



E4>t I— - e.^d 



sin(6) 



0.006 
0.005 
0.005 
0.004 
0.004 
E(>ty(G)0.003 
0.002 
0.002 
0.001 
0.001 





1 






































' v. 






















) 






















J 






















/ 










n 


/ 








x" 


s 








/)/ 


S 


s 


















lu 


tS 








__-- 












f// 


.- 




















//~ 



















-0.005 -0.002 0.002 0.006 0.009 0.013 0.016 0.02 0.023 0.027 0.0? 

E$tx(6) 

— E FIELD 



204 



Elevation Plot of Total Electric Field 



Et(9) : = J( |E0t(G,*g)| ) 2 + ( |E<frt(9,*g)| 



Etx(9) •= 



t|^-e 



•cos(G) Ety(6) •= 



:tl--e 



sin(9) 



0.006 



0.005 



0.004 



0.004 



0.00? 



Ety( 9) 0.003 



0.002 



0.002 



0.001 



0.001 





f 






























/ 
















t 




















1 


















/ 


/ 


















/ 
















\i 


// 


















if 

i, 


/ 


















If 






/ 














I 

i 
i — 


/ 


s 


/ 













-0.005-0.002 0.002 0.006 0.009 0.013 0.016 0.02 0.023 0.027 0.03 

Etx(8) 

— EFELD 



205 



THE SPIRAL ANTENNA (EXACT METHOD) 
MATHCAD SOFTWARE-SPIRAL. MCD 

Spiral antennas are a family of two or three dimensional devices that 
possess frequency independant parameters over a wide bandwidth. Spiral 
antennas are commonly used for direction finding, satellite tracking 
and missile guidance. 

The planar spiral may be of the Archemedean, log-spiral, or equiangular 
type. All three radiate two main, circularly polarized lobes 
perpendicular to the plane of the antenna. Additional gain for planar 
spirals may be achieved by placing a metal cavity on the side of the 
antenna with the unwanted lobe. The cavity may be empty or be filled 
with electromagnetic energy absorbing material. These applications 
principly examine the equiangular planar spiral and do not account for 
cavity backed effects. 

The three dimensional, or conical, spiral exhibits many of the same 
features as the planar spiral except that it radiates a single main beam 
in the direction of its tip, thereby eliminating the need for cavities. 
The conical log-spiral is the only three dimensional antenna analyzed by 
this application. 

(Note: Mathcad equations cannot use symbolic subscripts. Therefore, 
symbols like X will immediately follow the parameter in equations in 
lieu of subscripts. ) 

The spiral antenna Mathcad applications will compute the following 
parameters for equiangular planar spirals and conical log-spirals: 

k = Wavenumber 
). = Wavelength 
D Q = Directivity 

Eg = Electric Field (8) Component 

Ea = Electric Field ($) Component 

A,B,C = Conical Log-Spiral Electric Field Coefficients 

U = Radiation Intensity 

u max = Maximum Radiation Intensity 

p rad = Radiated Power 

G = Gain 

EIRP = Effective Isotropic Radiated Power 

A ern = Maximum Effective Aperture 

BW = Bandwidth 

r min = Minimum Distance to Far-Field 
R r = Radiation Resistance 
h em = Maximum Effective Height 

E x,y,z = Conical Log-Spiral Electric Field Spatial Components 
o a = Conical Log-Spiral Unit Polarization Vector 
^high = Upper Operating Frequency 
^low = Lower Operating Frequency 
r n = Any point on the n th edge of a spiral 
£ ex = Equiangular Planar Spiral Expansion Ratio 
Zi = Planar Spiral Input Impedance 
T = Voltage Reflection Coefficient 

206 



e rv = Reflection Efficiency 

PLF = Polarization Loss Factor 

Q = Conical Log-Spiral Antenna Slowness Factor 

I (0 = Conical Log-Spiral Current Distribution 

^(£) = Conical Log-Spiral Azimuth 

a n = Conical Log-Spiral Phase Difference of n th arm 

Xhiqh = Upper Operating Wavelength 

^low = Lower Operating Wavelength 

P = Planar Spiral Angle 

A = Planar Spiral Electric Field Amplitude 

b = Conical Log-Spiral Constant 

L = Conical Log-Spiral Total Arm Length 

( L = Conical Log-Spiral Azimuth at End of Arm 

8p = Desired Conical Log-Spiral Polarization Offset Angle from z Axis 
$p = Desired Conical Log-Spiral Polarization Azimuth Angle 

L = Conical Log-Spiral Total Arm Length 

The following data must be input based on known or estimated data: 

M = Mode 

N = Number of Spiral Arms 

f = Frequency of Interest 

i = Number of Increments for Far Field Radiation Patterns 

r^f = Distance of Far-Field Calculations 

I Q = Input Current at Antenna Terminals 

0q = Deflection Angle from +z Axis for Azimuth Plot 

(jiq = Azimuth Angle for Elevation Plot 

(x,y, z) = Coordinates for Conical Log-Spiral Unit Polarization Vector 
r Q = Spiral Feed Point 

a = Flare Rate 

8 n+ ]_ = Angular Arm Width of n^ Spiral Arm 

^ex = Azimuth to Compute Expansion Ratio 

P = Conical Log-Spiral Angle 

R = Overall Radius 

E Q = Source Strength Constant for Planar Spirals 

Zj_ = Conical Log-Spiral Input Impedance 

o w = Wave Unit Polarization Vector 

O = Cone Angle 

A9 = Conical Log-Spiral Half-Power Beamwidth 
Z c = Characteristic Impedance of Feed Assembly 

t = Number of Increments Along Conical Log-Spiral Arm 
o a = Equiangular Planar Spiral Unit Polarization Vector 



207 



THE PLANAR SPIRAL ANTENNAS 



Enter input data here 



N = 2 (arms) 

M := 1 (mode) 

(Note: M max is N-l) 

i = 30 (increments) 

a =.221 (dimensionlessi 

R = 1 (m) 



5 = 



71 

2 

71 

3-n 

2 



(radians 



f =3io° (Hz: 



Io := 1 (amps) 



rff=l 10 (meters) 



ro=.l 



m 



Eo =10" (V/m) 



j 





(dimensionless! 



Zo =100 (Q) 



Calculate planar spiral antenna geometric parameters and define 
constants : 



c =2. 9979 10 (meters/sec) 



TU -=120-7i 



(Q) 



I = 



(meters/cycle) 



- ] in" 9 

36 7t 



Farads/m) 



X = 99.93 (meters/cycle) 



H =4 Tt 10 (H/m) 



oa 



(dimensionless) 



208 



Calculate planar spiral antenna parameters : 



Define angular offset 
from y-z axis: 



Distance to Far-Field r 



min-i 



71 71 71 71 

2\ 11 



+ =0,^..2.a 

i 



(radians) 



(radians; 



rr -=1.6-X 



n-j =10R 



ir„ 



8-R' 



(m) 

(IB) 

(m) 



Wavenumber k 



rmin =max(rr) (m) 



rmin = 1.59888- 10 (m) 



k - 



2-71 



(nT 1 ) 



k = 0.06288 



(m 



-1 



Radial Distance to n t ^ 1 Spiral Edge r 



r(n,4>r) =roe 



\*-K. 



(m) 



r(l ; 2-7t) =0.40092 



(IB) 



Expansion Ratio £ py : 



eex(n^r) := 



r(n,i(ir4- 2-n) 
r(n^r) 



(dimensionless) 



eex( 1, 2- ji) =4.00917 



(dimensionless ) 



209 



Bandwidth BW: 



Equiangular Spiral 



Log-Periodic Spiral 



J.high =4ro 



(m) 



Xhighl =20ro 



(m) 



fhigh 



Xhigh 



fhigh =7.49475-10 



(Hz) 



(Hz; 



fhighl := 



Xhighl 

fhigh= 7.49475- 10 8 



(Hz) 



(Hz: 



Xlow MR 



(m) 



flow :=■ 



Mow 



(Hz) 



flow =7.49475- 10 



(Hz) 



BW = fhigh - flow 



(Hz) 



BW1 = fliighl-flow 



(Hz) 



BW= 6.74528- 10* 



(Hz) BW1 =7.49475-10 



(Hz) 



Electric Field E ( 6j ) and Electric Field Amplitude A(8): 



w:=0..i 



(increments) 



E^(iG) := 



M j 



- , - 1 r /i 

Eok'cos(8)(l + j -acos(9)) tani — | 

2! 

sin(9) z rff 



M- e+-|-k-rff 



% .j -M-+ 



(V/m) 



Ef -,-| =5.97188-10 5 +24182-10 4 j 

\6 6} 



(V/m) 



210 



Electric Field Amplitude A: 



A(6) = 



cos(0)tan|- -e^ a 
2 



2 2 

sin(0)-\'l + a cos(0) 



(V/m) 



Al 



+ 71 

7i w\ 2 i i 
cosi- — + 7i— -tan! -e 

2 i 2 



iM 

'M\ ... J. 



atan;a-cos \-% — 



71 W . 2 71 W 

sinj-— + 7i— J • 1 + a -cos!-- +■ n— 
2 i] ^ \ 2 i 



(V/m) 



Radiation Intensity U (0) 



uc©) =_L.(a(0) 2 ; 

2-Hn 



(W / solid angle! 



Ul 



« 2 . 



% 



Al 



(W / solid angle) 



Umax = max(Ul) 



(W / solid angle) 



Umax =2.2285*10 



"3 



(W / solid angle] 



Radiated Power P ra n : 



Prad :=4-» 



U(0) sin(0)d0 



(W) 



Prad =7.94528- 10 



-3 



211 



(W) 



Directivity D n : 



Do .= 



4-jt Umax 
Prad 



(dimensionless) 



Do =3.52462 



(dimensionless) 



Radiation Resistance Rr: 



Rr = 



2 Prad 
(ilo|) 2 



(fi) 



Rr = 0.01589 



(Q) 



Input Impedance Zi 



Zi : = 



N-30- 

. / M\ 
sin; n- — | 



(Q) 



Zi = 1.88496- 10" 



(0) 



Voltage Reflection Coefficient Y: 



r := 



Zi- Zo 
Zi-t- Zo 



(dimensionless! 



f= 0.30675 



(dimensionless) 



Reflecton Efficiency e r ^: 



erv:=l- (|ri) 2 



(dimensionless ) 



en- =0.90591 



212 



(dimensionless) 



G =ervDo 



G =3.19298 



EIRP : = Prad-Do 



(dimensionless) 



(dimensionless: 



GdB = 101og(G) (dBI 



GdB =5.04196 (dB) 



Effective Isotropic Radiated Power (EIRP) : 



(W) 



EIRP =0.028 (W) 



Polarization Loss Factor (PLF) 



PLF '= Ucwaa ) (dimensionless 



PLF = 1 (dimensionless; 



Maximum Effective Aperture (A f 



X 2 -Do-ctv-PLF . 2 s 

Aem = (m^) 

4 71 



Aem = 2.53733 -10 3 (m 2 ) 



213 



Maximum Effective Height (h PTn ) 



hem 



Rr-Aem 



(m) 



hem =0.65407 



(m) 



214 



THE EQUIANGULAR PLANAR SPIRAL ANTENNA FAR-FIELD ELEVATION PATTERN 



For the purpose of this far-field radiation pattern, the spiral antenna 
lies parallel to the Ey = grid line and is centered at the origin. 
The magitude of the electric field pattern is rotationally symmetric 
with respect to the Ex=0 grid line. The equiangular planar spiral 
antenna possesses a mirror image radiaton pattern in the -y half plane. 



E$x(8) ■= |A(6)|-cos 8 + 



Efo-(0) = |AC0)| -sinfe-t- — 



1.4 



1.26 



1.12 



0.98 



0.84 



E<t>y(6) 0.7 



0.56 



0.42 



0.28 



0.14 



-0.6 -0.48 -0.36 

— E FIELD 



-0.24 



-0.12 -2.776*10 

E4x(6) 



\T 






0.12 



0.24 



0.36 



0.48 



0.6 



215 



TEE CONICAL SPIRAL ANTENNAS 



Enter input data here 



N =2 (arms) 

M = 1 (mode) 

(Note: M max is N-l) 

i =9 (increments! 



f=280 10 (Hz) 

Io = 1 (amps) 

rflF = M0 4 (meters) 

ro =.03 (m) 



R = 15 (m) 



Zi =160 (Q) 



5 ■ = 



3* 



(radians ) 



o\v 



6 



(dimensionless; 



=73 (degrees! 



Zo =100 



(Q) 



180 



(radians 



6o = 10 (degrees; 



1.27409 (radians) 



x : = 



y =0 



z = 10 H 



m 



(m) 



(m) 



0o =0o 



180 



(radians) 



00 = 0.17453 (radians) 



HPBW =80 (degrees; 



216 



Calculate conical log-spiral antenna geometric parameters and define 
constants : 



c:=2. 997910 (meters/sec) 



n -=120-11 



(Q) 



(meters/cycle) 



1 9 

£ = 10 (Farads/m) 

36 7i 



31 = 1.07068 (meters/cycle) 



H =4-n-10 



-7 



(H/m) 



Calculate conical log-spiral antenna parameters 



Define angular offset 6 
from y-z axis: 



Distance to Far-Field rmin: 



n n n n 

2'i 2"2 



(radians 



rr :=1.6X 



n-j =10R 



(m) 



(m) 



rr„ 



8-R' 



(m) 



0, — .. 2-7i 



(radians) 



rmin =max(rr) (m) 



rmin = 1.71309 (m) 



Wavenumber k: 



Radial Distance to n *J l Spiral Edge r 



k : = 



2-71 



(m -1 ) 



b =cot(p) 



(dimensionless ) 



k = 5.86841 (rrT 1 ) 



b =0.30573 



(dimensionless) 



r(n,$r) = roe 



b-»in( Go)- (*-«„+]) (m) 



r(l,4-7c) =0.04948 
217 



(m) 



Bandwidth BW: 



Conical Log-Spiral 



Xhigh = 4ro 



(m) 



fhigh 



Xhigh 



(Hz) 



fhigh = 2.49825- 1CT 



(Hz; 



Mow =— R 
3 



(m) 



flow :=■ 



How 



(Hz 



flow =7.49475* 10 



(Hz) 



BW = fhjgh - flow 



(Hz) 



Antenna Slowness Factor Q: 



i b 



(dimensionless; 



Q =3.4203 



(dimensionless) 



Azimuth at End of the First Spiral Arm ALj 



R 



it 1 i sin(8o) 

<t>L=- -in' 



b-sin(Go) \ ro 



(radians) 



*L = 63.29231 



(radians; 



218 



Spiral Arm Length L: 



L =™- e * L - b ' sin(9o) -i; (m , 



L =2.72729 (m) 



Spiral Arm Current I ( £) 



t =25 



(increments; 



$:=0,-.L (m) 

t 



2 

t/cx -t L (amps; 

1(1) =Ioe 



Electric Field Coefficients g , A ( £ ) , B ( £ ) , C ( £ ) , D ( £ ) , & (£) 

2-71 

g := — 
N 



(dimensionless; 



o = 3. 14159 



(dimensionless; 



1 =o..N-l (increments! 



v =0..— (increments; 



w=0..i (increments) 



)1(0 = ■ln[i^+ 1 

bsin(9o) \ro / (radians) 



C6(U,D := 1 + _J W <♦«*>-♦+">+[] _ _J \. e -J <Wtt-*+H 

b-sin(9o)/ \ b-sin(0o)/ 

v " K ' (dimensionless) 

219 



C16(E,w,l) := 1 + 



J 



j ■ ♦l(^)-i-2-x + l-o 



b-sin(Oo)/ 



1- 



J 



bsin(0o) 



-j ■(♦!($)- T'2-*+l-a] 



(dimensionless) 



C*(^,*,l): = 



j J V gi •(♦K4)-* + l-o)_ L j \ e -j •(♦K5)-*+l-o)l ■ sin(6o) 



b-sin(Go) 



b-sin(Go) 



(dimensionless) 



Cl^,w,l) := 



1+— J -e l 

\ bsin(6o)/ 



1 - 



bsin(6o) 



• \ -j ■(♦!($)- *-2-x+l-a 



sin(9o) 



(dimensionless) 



A(E,e,*) :=£V j 

1 



■k-l 



Mia Q 
e x 



( sin( Ojsin( Go)cos( <J) 1 < £, > — $ + !•«)) 



sin(eo)cos(0)Ce(^il) . , Q . .. ,\ 
— - — — ^- L - 1 — -- sin(6)cos(6o); 



(dimensionless) 



Dl(^,v,w) 



/ Iv 71' 

sin(eo)-cos'-5i+--)-C10(^\v,l) 

! ■ sin— n+ -- -cos(0o) 

\ 2 \i 2/ 



(dimensionless; 



Altf.v.w) :=][Y j Mla e 



■L^i.[sin(--ii+--)-Hn(Gb)-cos[*l(^)---2-*-(-l-a' 



Q \ \i 2y \ i 



•Dl(4,v,w) 



(dimensionless; 



B($,e,t) : = ]Te j M1 a e 



-^■sin( 9)sin( 8o)cos( 4> 1 < \) - 4 f la) 



(dimensionless) 



•C*C^,4,1) 



220 



ni/ , . . \ ' -j Mia 

Bl(!j,v,w) = y e 

1 



i-^sin(-x + --Vsin(eo)cos^l(0-^-2)t + la 



Q \i 2, 



Electric Field Compontents E 6, Ejj_ 



ci<KU,D 



(dimensionless ) 



rL 



E0(0,<t.) =-j fn r 



-j krff 

e J 
2-rffQ 



Kc)e Q 



i— ^5C0S(9)C0S(G0) 



■A(^,G,*)d; 



(V/m) 



E01 : = -j -f-ji. 



-j -krff 



2-rff-Q . 



i_^. C os(l.x+-*)-cos(8o) 

K^)e Q Vi 2/ Al(^v,w)d4 



(V/m) 



EK6,*) : = f-u 



-j -krff 

2-rff-Q . 



KO-e 



J— ^-cos(6)-cos(9o) 

Q 



Boue.tK 



(V/m) 



E4»l :=f-|L, 

1 v, w 



2-rff-Q "j 



i --JH ■ 



KO-e 



•cos I 1— ■* + ] i -cos( 6o) 

Q \\i 2// 



Bl(^,v,w)d4 







(V/m) 



221 



Radiation Intensity U( 8) 



U2(9,4>) :=—■[( |Ee(G»| )\ ( |E«M)| f] 

2^n 



(W / solid angle) 



U22 



rfi 2 r 



v,w ~ 



% 



JF.ei h 2 +(|E$i 



(W / solid angle! 



Umax =ma\(U22) 



(W / solid angle! 



Umax = 0.25969 



(W / solid angle; 



Radiated Power P rar ^: 



*2-7t Pti 



Prad 



U2(e,*)-sin(6)ded^ 



JO 



(W) 



Prad =0.81332 



(W) 



Directivity D : 



Do 



4-7i Umax 
Prad 



(dimensionless ) 



Do =4.01237 



(dimensionless) 



Ddbo:=10-log(Do) 



(dB) 



Ddbo = 6.03401 



(dB) 



Dldbo = 



32600 
HPBW 2 



(dB) 



Dldbo =5.09375 



(dB) 



222 



Voltage Reflection Coefficient T: 



Zi- Zo 
Zi+- Zo 



(dimensionless: 



T = 0.23077 



(dimensionless) 



Reflecton Efficiency e rv j 



etv =i-(irj) 2 



(dimensionless) 



erv =0.94675 



(dimensionless 



Gain G: 



G = erv- Do 



(dimensionless) 



G = 3.7987 



(dimensionless ) 



GdB =101og(G) 



(dB) 



GdB =5.79635 



(dB] 



Effective Isotropic Radiated Power (EIRP) : 



EIRP =PradDo 



(w; 



EIRP =3.26336 



(W) 



223 



Polarization Loss Factor PLF: 



0p - atan 



2 2 

x +y 



(radians) 



(fip =atan- 



(radians) 



Gp= 



(radians; 



<t»P = 



(radians) 



Ex:-E9(9p,(jtp)-cos(0p)cos((tip)- EK8p,fo>)-sin(c>p) (V/m) 



Ey =Ee(ep,(tip)cos(ep)sin(^p)- E(K6p,(()p)cos((t)p) (V/m) 



Ez :=-E6(0p^p)-sin(ep) 



(V/m) 



oa 



Ex\ 
Ey 
^(|Ex|) 2 + (|Eyj) 2 + (|Ez!) 2 \Ez/ 



(dimensionless] 



/-0.92973 - 0.1613J 
oa = 0.29553 + 0.14919J 





(dimensionless) 



= \ | o\v- oa ; ) 



(dimensionless) 



PLF = 0.40896 



(dimensionless) 



Maximum Effective Aperture (A PTr ) : 



Aem - 



X -Do-gv-PLF 
4-ji 



(m 2 ) 



Aem =0.14172 



(m 2 ) 



Maximum Effective Height (h em ) 



hem 



Rr Aem 



(m) 



hem = 4.88816* 10 



,-3 



224 



(m) 



THE CONICAL LOG-SPIRAL ANTENNA FAR-FIELD ELEVATION PATTERNS 



For the purpose of the far-field radiation elevation patterns, the 
conical log-spiral antenna lies parallel to the Ey = grid line. 
Application users must specify the desired azimuth and elevation for the 
plots . 

THETA COMPONENT OF THE ELEVATION PATTERN 



$g: = — (radians) 
2 



Ex(8) := |E8(e+ji,te)|cos 8 + 



Ey(6) :=|E8(e+ii,*g)|-sin(e + 



2: 



8.8*10 



Ey(G) 7*10 5 



5.2M0 5 



3.4M0 



1.6*10 5 



-2*10 6 



-2*10 5 



-8*10 5 -4*10 5 -1.355*10 _20 4*10 5 8*10 5 

Ex(G) 



E FIELD 



225 



PHI COMPONENT OF THE ELEVATION PATTERN 



4>g :=— (radians) 



Exl(G) :=|E4Ce + «,4g)|-cos + - 

\ 2 



Eyl(8) = |E4(e-i-7i,4g)|-sin 9 + 



Eyl(0) 



1*10 



5M0 5 



-5M0 -5 



5 p 


/ 






























/ 


f 














\ 


\ 


/ 

\ 


















1 




















\ 


\ 




















\ 


















1 


\ 


















1 

1 






\ 


l\ 






J> 




/ 
















































— E FIELD 



-8M0 5 -2.711M0 20 8M0 5 
Exl(G) 







226 



TOTAL ELEVATION PATTERN 



$g '-- (radians) 
2 



Etce.^-VEece.^+E^e,^) 2 



Ex2(6) := |Et(8+Mg)|-cos[9 + - 



Ey2(0)-=|Et(e+n,ig)|-sin(e+- 

\ 2 



0.001 



Ey2(G) 



6*10 5 



5M0 6 



-5*10 



-5 





























/ 












A 


\ 






















1 




\ 

\ 














1 




















I 






\ 




















\ 














1 






\ 


\ 












/ 

f 


































































-0.001 

— E FIELD 



-1*10 4 -2.7nM0 _20 lM0 4 
Ex2(6) 



0.001 



227 



THE CONICAL LOG-SPIRAL ANTENNA FAR-FIELD AZIMUTH PATTERNS 

For the purpose of the far-field radiation azimuth patterns, the conical 

log-spiral antenna lies in the plane of the plot and is centered at the 

origin. Application users must specify the desired offset angle from 
the z axis for the plots. 

THETA COMPONENT OF THE AZIMUTH PATTERN 
0g :=— (radians) 



Ex(*) = EeC9g,4)|-cos(t) 



Ey(t)-=|E9(eg,t)|-sin(t) 



-5 



5.5M0 



-no -5 



Ey(<t>)-7.5M0 



-5 





y> 




/\ 








A 
















\ y^ 


^ 


' 


\ 
















- 




























\ .0 
















































| 
























































\ 






















X 


\ 


1 


















\ 


J 



-0.001 

— EFELD 



4 -2.71 1M0 -20 1*10 4 



0.001 



Ex(*) 



228 



'j. inu j-vij xriu x 11 cm x uimi 



Exl(*):=|BK8g,4)|-oos(*) 



Eyl(*):=|EKeg,*)|-sin(A) 



0.001 



1M0 4 



Eyl(<ji) -2.71 1*10 



^20 



-1*10 4 



-o.ooi 









/ 










~~~~J 






s 




















1 










/ 


/ 




























) 






/ 













/ 


/ 






\ 


\ 





























































































— E FIELD 



"3*10 5 6*10 5 




Exl(*) 



0.001 0.001 



229 



TOTAL AZIMUTH PATTERN 



Ex2(*) =|Et(e g) *)jcos(*) 



Ey2(*):=|Et(9g,*)|-sin(t) 



0.001 



1M0 4 



Ey2(*) -2.711*10 20 



-1*10 



-0.001 











/ 


























/ 


/ 

/ 


















' 
















/ 

( 
















1 


















1 





























\ 













































































-o.ooi 

— E FIELD 



-2.711*10 
Ex2(*) 



"20 



0.001 



230 



THE CONICAL HORN ANTENNA (EXACT METHOD) 
MATHCAD SOFTWARE-HORN_CON.MCD 

Conical horn antennas are devices used to provide a transistion from a 
circular waveguide to an unbounded medium such that the wavefront at the 
aperture of the horn has nearly a constant phase at any point in the 
mouth of the horn. 

Conical horns are commonly used as feed elements for reflectors used in 
satellite tracking, microwave communications, and radar. 

(Note: Mathcad equations cannot use symbolic subscripts. Therefore, 
symbols like X will immediately follow the parameter in equations in 
lieu of subscripts.) 

The conical horn antenna Mathcad applications will compute the following 
parameters : 

k = Wavenumber 
X - Wavelength 
D = Directivity 

Eg = Electric Field (6) Component 
Ea = Electric Field (<|i) Component 
U = Radiation Intensity 
U max = Maximum Radiation Intensity 

P rac j = Radiated Power 

G = Gain 

EIRP = Effective Isotropic Radiated Power 

A em = Maximum Effective Aperture 

BW = Bandwidth 

r min = Minimum Distance to Far-Field 

R r = Radiation Resistance 

h em = Maximum Effective Height 

E x v i = Electric Field Components in Cartesian Coordinates 

o a = Unit Polarization Vector 

f cte = Transverse Electric Cutoff Frequencies 

f ctm = Transverse Magnetic Cutoff Frequencies 

e ap = Aperture Efficiency 

PLF = Polarization Loss Factor 

p v (cos(6)) = Associated Legendre Function of the First Kind 

H (2) v ,H (2) V '= Spherical Hankel Function and its Derivative 

v = Legendre and Hankel Function Order 
b ,8 = Legendre and Hankel Function Constants 
d optimum = Optimum Conical Horn Mouth Diameter 
A x y z = Conical Horn Magnetic Vector Potentials 

h = Conical Horn Axial Height 

B H = Magnetic Vector Potential Integral Coefficients 

P = Magnetic Vector Potential Integral Phase Shift 



231 



The following data must be input based on known or estimated data: 

m, n = Mode 

f = Frequency of Interest 

i = Number of Increments for Far Field Radiation Patterns 

tff = Distance of Far-Field Calculations 

I Q = Input Current at Antenna Terminals 

0q = Coaltitude (Deflection Angle from +z Axis) for Azimuth Plot 
$ q = Azimuth Angle for Elevation Plot 
(x,y,z) = Coordinates for Unit Polarization Vector 
a = Flare Angle 
a = Circular Waveguide Inner Radius 

Xm n = n*-* 1 Zero of Bessel Function of the First Kind, Order m. 

X 'm n = nt ^ Zero of Bessel Function Derivative of the First Kind, Order 

m. 

B Q = Electric Field Amplitude Constant 

d meas = Measured Diameter of the Conical Horn's Mouth 



Enter input data here: 



n =1 



(mode number) 



f =1.96 l(f (Hz: 



m =1 



(mode number; 



Io = 1 



( amp s ) 



i = li 



^increments ) 



rff = 110 (meters; 



a =.0445 (m) 



a =2— 



radians) 



G\V 



J_ 
j_ 

'■& 





(dimensionless) 



(Note: a crust be less than n/3) 



x=10 (m) 



Bo =1 



(V/m) 



y =10 (m) 



dmeas = 1 



(m) 



z =10 (m) 



232 



Calculate conical horn antenna geometric parameters and define 
constants : 



c =2.9979 10 (meters/sec) 



% -=120-ii 



(fi) 



(meters/cycle) 



10 



-9 



36 * 



(Farads/m) 



X =0.15295 (meters/cycle) 



M- =4 7i 10" 7 (H/m) 



doptimum = — X- 

2 . M 



sin 



W 



(m) 



dmeas 



h = 



atanl — 



(m) 



doptimum =0.67081 



(m; 



h= 1.48879 



(m) 



Define Bessel Function Matrices: 

(Note: m = column number, n = row number, matrix index starts at 
n = m =0. The n = row has no physical significance, it is 
only a placeholder.) 





2.4049 3.8318 5.1357 
5.5201 7.1056 8.4173 





6.3802 7.5884 
9.7610 11.0647 



8.6537 10.1735 11.6199 13.0152 14.3726 
11.7915 13.3227 14.7960 16.2235 17.6160 



(dimensionless) 



ZP 





3.8318 1.8412 

7.0156 5.3315 

10.1735 8.5363 9.9695 11.3459 12.6819 

13.3237 11.7060 13.1704 14.5859 15.9641 












3.0542 


4.2012 


5.3175 


6.7062 


8.0153 


9.2824 



(dimensionless ) 



233 



Calculate planar spiral antenna parameters : 

Distance to Far-Field r 



Define angular offset 6 
from y-z axis: 



Gp 



a a a a 
2'i 21 



$p =o,— ..2-Ti 
i 



(radians) 



(radians 



itu.ru. 



rr =1.6X 



rr = 5dmeas 



rr^ 



2dmeas^ 



(m) 



(m) 



(m) 



71 71 71 71 
2'i 2" 2 



(radians) 



rmin = max(rr) (i 



♦ =0,^..2n 
i 



(radians) 



rmin = 13.07582 (m) 



Wavenumber k: 



2-71 



(m _1 ) 



k = 41.0789 



(rrT 1 ) 



Cutoff Frequencies f r : 



Transverse Electric (TE) Modes 



fete 



ZP 



2*a> e 



(Hz: 



fcte = 



4.11 135-10 

7.52742- 10 9 



1.97552- 10 9 3.27702- 10 9 
5.72046-10 9 7.19545-10 9 



4.5077- 10' 5.70544-10 
8.60005- 10 9 9.95959- 10 9 



1.09157- 10 10 9.15906-10 9 1.06968-10 10 1.21736-10 10 1.36071-10 10 



jo 



.10 



10 



1.42957- 10 1 " 1.256-10'" 1.41312-10 1 " 1.565-10 



jo 



1.71288-10 



10 



:hz: 



234 



Transverse Magnetic (TM) Modes 



fctm = S- 



2 -"' a fo' E o 



(Hz) 



fctm = 



2.58035-10 9 4.11135-10 9 5.51037-10 9 6.84567-10 9 8.1420M0 9 

5.92282-10 9 7.62399-10 9 9.03138-10 9 1.04731-10 10 1.18719-10 10 ( 

9.28503-10 9 1.09157-10 10 1.24676-10 10 1.39647-10 10 1.5421 MO 10 

1.26517-10 10 1.42947-10 10 1.58754-10 10 1.74071 MO 10 1.89012-10 10 



iz: 



Legendre and Hankel Fuction Constants b ,5: 



«\ 



log 



TJ^T 



(dimensionless 



log COS; — 

\2 



b = 26.32708 



(dimensionless ) 



5=* 



(dimensionless) 



8 = 4.5 



(dimensionless ) 



Legendre and Hankel Function Orders y: 



v:=-.5+.5- 1+4-b 



(dimensionless ) 



v = 4.6553 



(dimensionless) 



235 



Legendre Fuctioas P v cos ( 8 ) : 



pl(9) •= F(v+2) ■ -•sin(9)-cos(v+.5)-e+- 
r(v+1.5)^ ! 2 4 



(dimensionless; 



pl(- =-1.51294 



(dimensionless; 



Hankel Function and its Derivative H ^)^ H ^ 2 \^_ 



kh 



(dimensionless ) 



-jf h -i*) / . v+ i 

Hv2p = — e '■ 1-j ■ 

kh kh 



(dimensionless; 



Magnetic Vector Potential Integral Coefficient Bi 



BH := ^ -e-J krff fi-Hv2 + k Hv2 

j (2-7i) 2 f-£ rff 



' 



(Wb/m) 



Magnetic Vector Potential Integral Phase Shift ^_ and Other Coefficients; 



p(0J,ep,*p) =cos(e)cos(ep) + sin(9) sin(9p)cos($- fo>) 



(dimensionless) 



C(9p,$p) : = (sin(8 0p)cos($p) cos(9p) + Scos(S 9p)sin(4>p) sin(9p)j 



(dimensionless) 



D(0p»$P) = sin(8-9p)cos((J»p) sin(^p) cos(6p) - 8-cos(5-9p)-sin(^p)-cos(^p)-sin(9p) 

236 



(dimensionless) 



Magnetic Vector Potential Integrals : 



A\(G,$) :=-BHpl|- 



a 

2 

- 



P2-ic 



e" khP(e '*' ep ^ ) C(0p > ip)%dep (Wb/m) 



Ay(G» : = -BH-pl 



U; j 



■2-n 



i kh 



p(e '*' ep '* p) -D(0p,ip)%de P 



(Wb/m) 



Az(G^) = -BHpl 



W 

2/ J 



a 

2 f 



J 



e" ' kh P(6 '*' 6p '* p) sin(6Gp)cos(^p)sin(Gp) % dGp 



(Wb/m) 



Electric Field Components E q ,E i: 



E9(G,4») - = k •(Ax(0,*)-cos(G)-cos(^) + Ay(9J)-cos(0)-sin(^)- Az(0,$)sin(6)) 



(V/m) 



EK6J) = -Ax(6,<fr)sin($) + Ay(0,4i)-cos(4>) 



(V/m) 



Radiation Intensity U( 6) 



U(8,*) :=—[( |E9(G,*)| ) 2 + ( |E*(M)| ) 2 ] 
2 % 



(W/solid angle 



237 



Radiated Power P ra d : 



P2-* 



Prad 



U(e,*)sin(6)ded4 



(W) 



Prad =8.23092-10' 



(W) 



Directivity D : 



Do 



4-7iU(0,0) 
Prad 



(dimensionless) 



Do =2.33662-10' 



(dimensionless ] 



Radiation Resistance R, 



Rr 



2 Prad 
(|lo|) 2 



(Q) 



Rr = 1.64618- 10" 



(Q) 



Gain G: 



G =.95 Do 



(dimensionless) 



G =2.21979-10' 



(dimensionless; 



GdB -=10-log(G) 



(dB) 



GdB =23.46311 



(dB; 



238 



Effective Isotropic Radiated Power EIRP: 



EIRP =PradDo 



(W) 



EIRP = 1.92325- 10' 



(W) 



Antenna Unit Polarization Vector gp, 



0pl = atan 



2 2 

x +y 



(radians; 



$pl = atan - 



(radians) 



9pl =0.95532 



(radians) ^pl =0.7854 



(radians) 



Ex :=E6(0pl,4pl)-cos(epl)-cos(^pl)- EK9pl,fo>l)-sin(fo)l) 



(V/m) 



Ey =E9(eplJpl)cos(epl)sin(^p]) + E(Kepl^pl)cos(^pl) 



(V/m) 



Ez =-E8(8pl,4»pl)-sin(epl) 



(V/m) 



oa :=■ 



(|Ex|) 2 + (|Ey|) 2 +(|Ez|) 2 



/Ex'~ 
Ey 
\Ez 



(dimensionless ) 



oa = 



0.05346 + 0.40449j 

0.05361 + 0.40495J 

\-0. 10708- 0.8094 5j 



(dimensionless ) 



Polarization Loss Factor PLF: 



|\2 

PLF := 1 1 ow oa 



(dimensionless ) 



PLF =0.16663 



(dimensionless) 



239 



Maximum Effective Aperture (A ^^) : 



X 2 -Do-.95-PLF . 2, 

Aem = (m^) 

4-71 



Aem = 0.06886 (m 2 ) 



Maximum Effective Height (h PTn ) 



hem := Rr-Aem -2 

4 % 



hem = 1.09672 



Aperture Efficiency e 



'dmeas 

71 



\2 



(m) 



(m) 



m- 



Aem ... . . 

eap = (dimensionless) 



eap =0.08768 (dimensionless! 



240 



THE CONICAL HORN ANTENNA FAR-FIELD ELEVATION PATTERNS 



For the purpose of these far-field radiation patterns, the conial horn 
antenna axis is parallel to the Ey = grid line and the apex of the 
horn is located at the origin. Electric field components behind the 
horn's aperture are assumed to be zero. 



Theta Component of Elevation Pattern 



4>g =— (radians) 

2 



E9x(e) := |E8(0,te)!-cos(e) 



E0y(0) := |Ee(e,te)j-sin(9) 



2.5*10 



2M0~* 



1.5*10 



1M0 -8 



5M0 -9 



E6y(9) -1.654*10 



-24 



c 




























































\ 

/ 
















































































\ 




















/ 










































-$., 


.-i6 , ,. 


,n"^ -*. 


,n-9 9 *. 


, n -9 ,<. 


-fl 

in ' < /;• 


in' /; 7« 


—6 

in * 7 fi» 


— o 
in * « o« 


i n 'i »in * 



-Q 

-5*10 



-1M0 -8 



1.5*10~* 



-2*10 



-2.5*10 

-1*10 f*10 *" 1.2*10 ' 2.3*10 ' 3.4*10 ' 

E6x(8) 
241 



E FIELD 



Phi Component of Elevation Pattern 



E4x(6) = |E4(0^g)|-cos(e) 



E*y(6) := |E*(9,*g)| -sin(9) 



0.002 



0.002 



0.001 



0.001 



EQ>( 6) -1.084*10 



-19 



-0.001 



-0.001 



-0.002 



-0.002 












y\ 




















y x 


\ 












1 


/ 

1 








\ 










1 












\ 


*« 




1 




















\ 


















/ 


























v 








/ 
















\ / 


/ 


















V 













-0.002 0.003 0.005 0.008 0.01 0.012 0.015 0.017 0.02 0.022 

E(trx(9) 

E FIELD 



242 



Total Elevation Pattern 



Et(9^):^E0(e,*) 2 + EKe» 2 



Etx(8):=|Et(Mg)|-cos(e) 



Ety(e) = |Et(G>g)|sin(G) 



0.002 



0.002 



0.001 



0.001 



Ety( 6) -1.084*10 



•in"" 



-o.ooi 



-o.ooi 



-0.002 



-0.002 



b 








y\ 




















/ \ 


\ 














r 






























\ 


\ 






■ 
















\ 


| 


\ 
















/ 
















/ 


/ 








\ 








/ 
















\ / 




















>• 













-0.002 0.003 

— E FIELD 



0.005 



0.008 0.01 

Etx(6) 



0.012 0.015 0.017 0.02 0.022 



243 



THE CONICAL HORN ANTENNA FAR-FIELD AZIMUTH PATTERNS 



For the purpose of these far-field radiation patterns, the conial horn 
antenna axis is perpendicular to the Ey = and Ex = grid lines and 
the apex of the horn is located at the origin. 

Theta Component of Azimuth Pattern 



6g = 



10 



(Note: 0g must be less than a/2 radians) 



E8x(4):=|E6(eg,4)|-cos(4) 



Eey(4)=|E6(6g»|sin(4) 



E9y(4i)-2.22'10 16 



-0.8 




-2.4 



-S -6.4 -4.8 -3.2 

— EFELD 



1.6 -4.441*10 16 1.6 
Efe(+) 

244 



3.2 4.8 



6.4 



Phi Component of Azimuth Pattern 



E*x(*):=|EKeg,*)|-eos(t) 



Efr(*):=|E«fc.*)|-an(*) 



0.005 



0.004 



0.003 



0.002 



0.001 



E4y(4) 



-o.ooi 



-0.002 



-0.003 



-fl.004 



-0.005 




-0.003 -0.002 -0.002 ~0.001 -fl.001 0.001 0.001 0.002 0.002 0.003 

E*(+) 
245 



E FIELD 



Total Azimuth Pattern 



Etce^^Ew^+Etfe,*) 2 



Etx(4):=|Et(eg,*)|-cos(*) 



Ety(4) :=|Et(9g,4)|-sin(4) 



Ety(^)-2.22M0 16 




-0.8 



-1.6 



-2.4 



-3.2 



246 



THE PYRAMIDAL HORN ANTENNA 
MATHCAD SOFTWARE-HORN_PYR.MCD 

Pyramidal horn antennas are devices used to provide a transistion from a 
rectangular waveguide to an unbounded medium such that the wavefront at 
the aperture of the horn has nearly a constant phase at any point in the 
mouth of the horn. 

Pyramidal horns are the most popular type of feed elements for 
reflectors used in satellite tracking, microwave communications, and 
radar. 

The pyramidal horn applications may be used to analyze E- and H-plane 
sectoral horns. To analyze an E-plane sectoral horn set horn dimension 
(al) equal to waveguide dimension (a) . To analyze an H-plane sectoral 
horn set horn dimension (bl) equal to waveguide dimension (b) . 

(Note: Mathcad equations cannot use symbolic subscripts. Therefore, 
symbols like X will immediately follow the parameter in equations in 
lieu of subscipts.) 

The pyramidal horn antenna Mathcad applications will compute the 
following parameters: 

k = Wavenumber 
X = Wavelength 
D Q = Directivity 

Eq = Electric Field (0) Component 

Ea = Electric Field ($) Component 

U = Radiation Intensity 

u max = Maximum Radiation Intensity 

p rad = Radiated Power 

G = Gain 

EIRP = Effective Isotropic Radiated Power 

A em = Maximum Effective Aperture 

BW = Bandwidth 

r min = Minimum Distance to Far-Field 

R r = Radiation Resistance 

h em = Maximum Effective Height 

E x v z = Electric Field Components in Cartesian Coordinates 

o a = Unit Polarization Vector 

f c = Transverse Electric and Transverse Magnetic Cutoff Frequencies 

e ap = Aperture Efficiency 

PLF = Polarization Loss Factor 

Dp = Pyramidal Horn Corner to Corner Distance 

Pe h = Pyramidal Horn Perpendicular Flare to Mouth Distances 

Ii 2 = Electric Field Component Equation Coefficients 



247 



The following data must be input based on known or estimated data: 

t = Number of Cutoff Frequencies Calculated 

m, n = Modes 

f = Frequency of Interest 

i = Number of Increments for Far Field Radiation Patterns 

r ff = Distance of Far-Field Calculations 

I Q = Input Current at Antenna Terminals 

9g = Coaltitude (Deflection Angle from +z Axis) for Azimuth Plot 

^ g = Azimuth Angle for Elevation Plot 

(x,y,z) = Coordinates for Unit Polarization Vector 
a,b = Rectangular Waveguide Dimensions 
a^/b^ = Pyramidal Horn Dimensions 

Pl/P2'Pe'Ph = Pyramidal Horn Imaginary Cone Apex to Mouth Distances 
E Q = Electric Field Amplitude Constant 



248 



Enter input data here: 



t:=5 



(modes) 



f = 9.310* (Hz) 



Io=l 



(amps) 



i =36 



(increments) 



rff = 110 (meters) 



(Note: For E-plane sectoral horn analysis 
set (al) equal to (a) . For H -plane sectoral 
horn analysis set (bl) equal to (a)) 



a:=.02286 (rn) 



al =.1846 (m) 



b =.01016 (m) 



bl =.1455 (m) 



' '■- J 

J 





pi =.3398 (m) 



(dimensionless) 



pe=.3281 
ph =.3521 

Eo=l 



(m) 



(m) 



(V/m) 



p2 =.3198 (m) 

x =10 3 (m) 

y = 10 3 (m) 
z = 10 



249 



Calculate pyramidal horn antenna geometric parameters and define 
constants : 



c =2.9979 10 8 (meters/sec) 



% 



= 120 7i 



(Q) 



X=5 



(meters/cycle) 



1 g 

e.:= 10 (Farads/m) 

36ti 



X = 0.03224 (meters/cycle ) 



(i :=4n-10 



,-i 



(H/m) 



: ^/a 



Dp=Val 2 +bl 2 



(m) 



Dp =0.23505 



(m) 



pe:=(b.-b).||H) 2 .I (m) 



p t :=(a,- a ,. | (ij 2 -i 



(m) 



pe = 0.29759 



(m) 



ph= 0.29771 



(m) 



Calculate planar spiral antenna parameters : 

Distance to Far-Field r 



Define angular offset 8 
from y-z axis: 



rr =1.6X 



minj. 



<ni) 



TIJl 71 ln -6 71 1A -6 
. = -— , +10 ..— + 10 

2 i 2 2 



(radians) 



it,- 5 Dp 



2-iy 



(m) 



(m) 



t =10- 6 ,— +10" 6 ..2n+10- 6 (radians) 



nnin =max(rr) (m) 

rmin = 3.42774 (m) 



250 



Wavenumber k: 



2-71 



(m" 1 ) 



k=1.94915-10 2 



(m" 1 ) 



Cutoff Frequencies f r : 



n=0..t 



(modes) 



m =0..t 



(modes) 




2 >o E o 



(HZ) 



fc = 



1.47638-10 10 2.95276-10 10 4.42913-10 10 5.90551-10 10 7.38189-10 10 

6.56168-10 9 1.61563-10 10 3.02478-10 10 4.47748-10 10 5.94185-10 10 7.41 MO 10 

1.31234'10 10 1.97533-10 10 3.23125-10 10 4.61946-10 10 6.04957«10 10 7.49763M0 10 

1.9685- 10 10 2.46063- 10 10 3.54877- 10 10 4.84688- 10 10 6.22496- 10 10 7.63985- l6 HE) 

2.62467- 10 10 3.01 141 -10 10 3.95065- 10 10 5. 14841 -10 10 6.46251 -10 10 7.83462- 10 10 

3.28084-10 10 3.59772-10 10 4.41392-10 10 5.51191-10 10 6.75566-10 10 8.07813-10 10 



(Note: The index for both m, n above begins with zero. TM modes cannot 
have m or n equal zero. TEqq mode does not physically exist.) 



251 



Electric Field Component Coefficients I -| ,I? : 



v=0..i 



(increments; 



w =0..i 



(increments) 



"(M) 



al 
2 



al 
2 



cos it-— -e 
\ 2/ 



-j -k U--.in(G)co«(*)^ 



2p2 



3 



(dimensionless ) 



II 



v,w 



cos 7t— -e 
\ 2/ 



2p2 \' 2, 



.j . k .|J .in(^*-*)-cos(^-2-*H 



<% 



al 
2 



(dimensionless ) 



ii(e,*):=if 



al=a,-7t — 
2 



cos k---sin(9)-cos(4>) 
2 



k-sin(G)cos(4>) 
2 



,11(6,*) 



(dimensionless) 



111 :=tf 

v,w 



al=a,-7t— • 
2 



coslk-sin — n — + 10" cos — 2-jh- 10" 



k^sm(^-^ + 10- 6 ].cos^-2-^10- 6 ]j -ll 



,111 



v,w 



(dimensionless) 



12(0,*) 



bl 

2 

bl 

2 



-j -k-U— -sin(e). s in(*)^ 



2pl 



d4 



(dimensionless ) 



252 



121 



v,w 



2 

bl 
2 



■ J *&-"M*(i*« 



&, 



(dimensionless ) 



I2(9,*):=if 



bl=b,b 



sin(k— sin(8)sin(4) 



k— sin(8) sin(4) 

2 



,12(6,4) 



(dimensionless) 



121 : = tf 

v,w 



bl=b,b 



sin | k--- sin --7i + 10" 1-sinf — -2-n+ 10" 

2 l 2 



\ i 



k^.sint,-l + 10- 6 |.sin(^2^10- 6 
2 i 2 



Electric Field Components Efl,Ei ,: 



,121 



v,w 



(dimensionless ) 



-j -k-rff 

E6(e,4) =j k-Eo- (sin(4)(l + cos(e)) 11(8,4)12(6,$)) 

4-n-rff 



(V/m) 



-j -k-iff 

E61 :=j -k-Eo-- 

4-ji-ra 



sinp.2.| 1+cos^J) ll VfW .Dl, 



(V/m) 



EtfM) =j k-Eo 



4-a-rff 



•(cos(4) (1 + cos(8)) 11(8,*) 12(6,*)) 



(V/m) 



-j kiff 

EA1 ;=j -k-Eo-- 

4«ra 



cos^.2.|(l + cos^-|]].Ill vw .I21 VfW 

253 



(V/m) 



Radiation Intensity U(8): 



U(M) :=—•[( |E9(e,4)| ) 2 + ( |EK6,*)| )' 
2 \ 



(W/solid angle) 



Ul 



iff 2 



v,w « 



\ 



(|E81 h 2 +(|E41 
\ I v . w l / \ I v »^ 



(W/solid angle) 



Umax = max(Ul) 



(W/solid angle) 



Radiated Power P 



racL. 



n-n 



Prad = 



Jo 



U(e,4)-sin(e)d9d4 



(W) 



Prad =3.39938-10 



-5 



(W) 



Directivity D n : 



Do = 



4 jt Umax 
Prad 



(dimensionless ) 



Do = 1.47304- \0 Z 



(dimensionless ) 



Radiation Resistance R 



r_L 



Rr = 



2 Prad 
(|lo|) 2 



(Q) 



Rr = 6.79875- 10 



-5 



(Q) 



254 



Gain G: 



G=.5Do 



(dimensionless) 



G =73.65177 



(dimensionless) 



GdB:=10-log(G) 



(dB) 



GdB = 18.67183 



(dB) 



Effective Isotropic Radiated Power EIRP: 



EIRP : = Prad-Do 



(w; 



EIRP =5.0074- 10 



"3 



(W) 



Antenna Unit Polarization Vector o a : 



6p =atan 



2 2 

x -i-y 



(radians) $P= atan 



(radians) 



9p =0.95532 



(radians) 



$p =0.7854 



(radians) 



Ex :=E8(ep,4p)-cos(8p)-cos(to>) - E4<ep,$p)sin($p) 



(V/m) 



Ey :=E9(0p,$p)-cos(0p)-sin(4ip) + E4<6p,$p)-cos(4>p) 



(V/m) 



Ez: = -E9(ep,4p)-sin(ep) 



(V/m) 



oa : = • 



JEx\ 
Ey 



^(|Ex|) 2 +(|Ey|) 2 + (|Ez|) 2 \Ez 



(dimensionless) 



/-0. 18825 + 0.09603j 
oa = 0.70255 - 0.35837J 
U-5143 + 0.26235J / 



255 



(dimensionless ) 



Polarization Loss Factor PLF: 



=-(l 



i\2 



PLF :=l orw-oa 



(dimensionless ] 



PLF =0.33333 



(dimensionless ) 



Maximum Effective Aperture A pm : 



Aem = 



X Do .5 PLF 

4-71 



(m 2 ) 



Aem =2.0301 1-10 



-3 



(m 2 ) 



Maximum Effective Height h ^ m : 



hem- RrAem -2 



(m) 



hem = 3.82683- 10 



-5 



(m) 



Aperture Efficiency e^: 



eap : = - 



Aem 
al-bl 



(dimensionless) 



eap =0.07558 



(dimensionless ) 



256 



THE PYRAMIDAL HORN ANTENNA FAR-FIELD ELEVATION PATTERNS 



For the purpose of these far-field radiation patterns, the pyramidal 
horn antenna central axis is parallel to the Ey = grid line and the 
perimeter of the mouth of the horn is parallel to the Ex = grid line. 
Electric field components in the half space behind the horn's aperture 
are assumed to be zero. 

r.l 



Theta Component of Elevation Pattern 



$g =0 (radians) 



E9x(6) := |E6(e,te)|-cos(e) 



Eey(0):=|E6(e,*g)| -sinCG) 



0.8 



0.6 



0.4 



0.2 



E9y(0) -5.551*10 17 



-0.2 



-0.4 



-0.6 



-0.8 



-1 



) 

- r 



-1 -0.8 "0.6 

— E FIELD 



-0.4 



-0.2 -5.551M0 17 0.2 

E9x(6) 

.on 257 



0.4 



0.6 



0.8 



Phi Component of Elevation Pattern 



E*x(6)-=|E*(e,*g)|-cos(e) 



E*y(e):=|E*(e>g)|-sin(0) 



0.001 



0.001 



E*y(6) 1.626* 10 19 



-0.001 



-0.001 



) 




K 


/ 




\ 












/ 


/ 


/ 






\ 








f 


1 












x 






f 














\ 


\ 






















\ 


I 


















} 


4 




















\ 


\ 




















\ 


\\ 






/ 














\ 


\ 




y 











0.001 0.001 

E FIELD 



0.002 0.002 0.003 

E4>x(9) 

258 



0.004 0.004 0.005 0.005 0.006 



Total Elevation Pattern 



Et(8,*) : = ^E8(8,t) 2 +E4(8,4) 2 



Etx(8) =(|Et(0,4g)!)cos(0) Ety(G) =(|Et(0^g)|)sin(0) 



0.001 



0.001 



Ety(6) 1.626*10 



nn-19 



-o.ooi 



-o.ooi 



3 




i 






N 












/ 


I / 
















//I 


/ 

/ 










X 


x 






f 




























































\ 
















/ 




\| 


\ 












/ 








\ 


j\ 


\ 






/ 












I 






/ 











0.001 0.001 0.002 0.002 0.003 0.004 0.004 0.005 0.005 0.006 

Etx(G) 



— E FIELD 



259 



THE PYRAMIDAL HORN ANTENNA FAR-FIELD AZIMUTH PATTERNS 



For the purpose of these far-field radiation patterns, the pyramidal 
horn central axis is perpendicular to the Ey = and Ex = grid lines 



Theta Component of Azimuth Pattern 



10 



(Note: Gg must be less than a/2 radians) 



E6x(*) = |E6(eg,*)|-cos(*) 



E0y(*) = |E9(Gg,4)|-sin(*) 



0.002 



0.002 



0.001 



0.001 



E9y($)~l-084M0 



"19 



-0.001 



-0.001 



-0.002 



-0.002 


























/ 












\ 


\ 
























{ 


















> 








































1 


( 


















I) 
























\ 














/ 

























-0.001 

E FIELD 



-2.71 1M0 -20 
E6x($) 







0.001 



260 



Phi Component of Azimuth Pattern 



E*x(t).= |E4(0g,*)|-cos($) 



E4y(4).= |Etfeg»|-sin(t) 



0.001 



El)y(if)-2J11'\0' 20 



-0.001 













) 












/ 






\ 


/ 












/ 






\ 


/ 












1 














\ 






' 














\ 






1 














\ 






i 

i 




















\ 




















\ 






/ 


\ 




s 


! 










■ \ 


/ 


\ 


y 









-0.002 -0.002 -0.001 -0.001 -1.084*10 0.001 0.001 002 0.002 



E FIELD 



E<px($) 



261 



Total Azimuth Pattern 



Et(e,4>) ^Ee(e,$) 2 + EKej) 2 



Etx(4):=|B(«g,4)| ■«*(♦) 



Ety(*) := |Et(Gg,*)|sin(*) 



0.002 



0.002 



0.001 



0.001 



Ety((Ji) "1.084*10 19 



-0.001 



-0.001 



-t).002 



-0.002 













) 




















\ 


\ 














1 






























X 






1 












































\ 










































\ 


J 































-0.002 -0.002 -0.001 -O.001 -1.084*10 19 0.001 0.001 0.002 0.002 



E FIELD 



Etx($) 



262 



REFERENCES 



1. Dietrich, D. S., "Predicting Radiation Characteristics from Antenna Physi- 
cal Dimensions," Naval Postgraduate School Technical Report NPSEC-93-009, 
1992. 

2. Kraus, J. D., Antennas, McGraw-Hill Book Company, 1988. 

3. Balanis, C. A., Antenna Theory Analysis and Design, John Wiley & Sons, Inc., 
1982. 

4. Cheng, D. K., Field and Wave Electromagnetics, Addison- Wesley Publishing 
Company, 1989. 

5. Stutzman, W\ L., and Thiele. G. A., Antenna Theory and Design, John Wiley 
&: Sons, Inc., 1981. 

6. Johnson, R. C, and Jasik, H., Antenna Engineering Handbook, 2nd ed., McGraw- 
Hill Book Company, 1984. 

7. Misek, V. A., The Beverage Antenna Handbook, V. A. Misek, publisher, 1987. 

8. Walter, C. H., Traveling Wave Antennas, McGraw-Hill Book Company, 1965. 

9. Belrose, J. S., Litva, J., Moss, G. E., and Stevens, E. E., "Beverage Antennas 
for Amateur Communications," QST, vol. 67, pp. 22-27, Jan. 1983. 

10. Jasik, H., Antenna Engineering Handbook, McGraw-Hill Book Company, 1961. 

11. Wolff, E. A., Antenna Analysis, John Wiley & Sons, 1966. 

12. Chang, S., and Maddocks, H. C, "APACK, A Combined Antenna and Propa- 
gation Model," ITT Research Institute, Technical Report ESD-TR-80-102, July 
1981. 

13. Lo, Y. T., and Lee, S. W., Antenna Handbook, Van Nostrand Reinhold Com- 
pany, 1988. 

14. Cheo, R. S., Rumsey, V. H., and Welch, W. J., "A Solution to the Frequency- 
Independent Antenna Problem," IRE Trans, on Antennas and Propagation, 
vol. AP-9, pp. 527-534, Nov. 1961. 

15. Corzine, R. G., and Mosko, J. A., Four Arm Spiral Antennas, Artech House 
Inc., 1990. 



263 



16. Dyson, J. D., "The Equiangular Spiral Antenna," IRE Trans, on Antennas and 
Propagation, vol. AP-7, pp. 181-187, Apr. 1959. 

17. Atia, A. E., and Mei, K. K., "Analysis of Multiple- Arm Conical Log-Spiral 
Antennas," IEEE Trans, on Antennas and Propagation, vol. AP-19, pp. 320- 
331, May 1971. 

18. Dyson, J. D., "The Unidirectional Equiangular Spiral Antenna," IRE Trans, 
on Antennas and Propagation, vol. AP-7, pp. 329-334, Oct. 1959. 

19. Balanis, C. A., Advanced Engineering Electromagnetics, John Wiley & Sons, 
Inc., 1989. 

20. Schorr, M. G., and Beck, Jr., F. J., "Electromagnetic Field of the Conical Horn," 
J. of Applied Physics, vol. 21, pp. 795-801, Aug. 1950. 

21. Hamid, M. A. K., "Diffraction by a Conical Horn," IEEE Trans, on Antennas 
and Propagation, vol. AP-26, pp. 520-528, Sep. 1968. 

22. Southworth, G. C, and King, A. P., "Metal Horns as Directive Receivers of 
Ultra-Short Waves," IRE Proc, vol. 27, pp. 95-102, Feb. 1939. 

23. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical 
Tables, M. Abramowitz and I. A. Stegun, eds., National Bureau of Standards, 
Dec. 1972. 



264 



INITIAL DISTRIBUTION LIST 



1. Defense Technical Information Center 
Cameron Station 

Alexandria, VA 22304-6145 

2. Library, Code 52 

Naval Postgraduate School 
Monterey, CA 93943-5002 

3. Chairman, Code EC 

Department of Electrical and Computer Engineering 
Naval Postgraduate School 
Monterey, CA 93943-5002 

4. Professor R. Clark Robertson, Code EC/Re 
Department of Electrical and Computer Engineering 
Naval Postgraduate School 

Monterey, CA 93943-5002 

5. Lieutenant Frank Kragh, Code EC/Kh 

Department of Electrical and Computer Engineering 
Naval Postgraduate School 
Monterey, CA 93943-5002 

6. LCDR Donald Gerry 
90 Jefferson Rd. 
Bourne, MA 02532 

7 . NAVMARINTCEN 

ATTN: Mr. Ron Ullom 

DI433 

4301 Suitland 

Washington D.C. 20395-5020 

8. Professor M.A. Morgan, Code EC/Mw 

Department of Electrical and Computer Engineering 
Naval Postgraduate School 
Monterey, CA 93943-5002 

9. Professor David Jenn, Code EC/Jn 

Department of Electrical and Computer Engineering 
Naval Postgraduate School 
Monterey, CA 93943-5002 



265 



DUDLEY KNOX LIBRARY 




3 2768 00331624 1